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"" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 261 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 261 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 261 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 663 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 665 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 666 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(21)" }{TEXT 403 1 " " }{TEXT 402 40 "Eigenvalues and e igenvectors of matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 739 10 "OBJECTIVES" }{TEXT 740 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 741 1 "\225" }{TEXT -1 50 " To provide a concise analysis of solution of a " }{TEXT 742 11 "homogeneous" }{TEXT -1 32 " system of algebra ic equations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 744 1 "\225" }{TEXT -1 36 " To introduce the concepts of the " }{TEXT 743 21 "characteristi c matrix" }{TEXT -1 3 ", " }{TEXT 745 26 "characteristic determinant " }{TEXT -1 3 ", " }{TEXT 746 25 "characteristic polynomial" }{TEXT -1 8 ", and " }{TEXT 747 23 "characteristic equation" }{TEXT -1 21 " of a square matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 750 1 "\225" } {TEXT -1 32 " To introduce the concepts of " }{TEXT 748 11 "eigenval ues" }{TEXT -1 6 " or " }{TEXT 749 20 "characteristic roots" }{TEXT -1 29 " of a square matrix and of " }{TEXT 751 12 "eigenvectors" } {TEXT -1 6 " or " }{TEXT 752 22 "characteristic vectors" }{TEXT -1 38 " corresponding to matrix eigenvalues." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 753 1 "\225" }{TEXT -1 114 " To provide a step-by-step example \+ of computing eigenvalues and eigenvectors of a matrix with numerical e lements." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 758 1 "\225" }{TEXT -1 28 " \+ To introduce the function " }{TEXT 757 10 "eigenvects" }{TEXT -1 52 " and analyse the structure returned by the function." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 764 1 "\225" }{TEXT -1 129 " To provide selection pro cedures for extracting in a unique way eigenvalues and eigenvectors fr om the structure returned by the " }{TEXT 765 10 "eigenvects" }{TEXT -1 10 " function." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 761 1 "\225" } {TEXT -1 35 " To introduce the concept of the " }{TEXT 759 9 "Froben ius" }{TEXT -1 2 " " }{TEXT 760 4 "norm" }{TEXT -1 76 " of a vector \+ and show how to use it for computing the norm of eigenvectors." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 763 1 "\225" }{TEXT -1 65 " To show how \+ to perform normalization of eigenvectors using the " }{TEXT 762 9 "nor malize" }{TEXT -1 10 " function." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 766 1 "\225" }{TEXT -1 71 " To specify and illustrate properties of eigen values and eigenvectors." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 768 1 "\225 " }{TEXT -1 15 " To give the " }{TEXT 767 15 "Cayley-Hamilton" } {TEXT -1 74 " theorem and provide a numerical example for verificatio n of the theorem." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 769 1 "\225" } {TEXT -1 36 " To suggest an application of the " }{TEXT 770 15 "Cayl ey-Hamilton" }{TEXT -1 45 " theorem to computing functions of matrice s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 158 "restart : interface(warnlevel=0) : with(linalg , charmat, charpoly, det, diag, eigenvals, eigenvects, inverse, norm, \+ normalize, trace, transpose, vector) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 672 3 "A. " }{TEXT 673 26 "Definitions and properties" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "It was stated in Unit (19) t hat a system of linear algebraic inhomogeneous equations may be writte n in the matrix form" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" } {TEXT 258 1 "A" }{TEXT -1 3 "] [" }{TEXT 259 1 "X" }{TEXT -1 1 "]" } {TEXT 595 3 " = " }{TEXT -1 1 "[" }{TEXT 260 1 "K" }{TEXT -1 1 "]" } {TEXT 578 75 " \+ (1)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where [" }{TEXT 261 1 "A" }{TEXT -1 9 "] is an " }{TEXT 596 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 262 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 597 1 ")" }{TEXT -1 2 " " }{TEXT 266 11 "coefficient" }{TEXT -1 11 " matrix, [" } {TEXT 263 1 "X" }{TEXT -1 8 "] is a " }{TEXT 598 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 264 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" } {TEXT 599 1 ")" }{TEXT -1 2 " " }{TEXT 265 8 "solution" }{TEXT -1 31 " column matrix (vector), and [" }{TEXT 267 1 "K" }{TEXT -1 8 "] is a " }{TEXT 600 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 268 3 " \327 \+ " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 601 1 ")" }{TEXT -1 2 " " } {TEXT 269 13 "inhomogeneous" }{TEXT -1 25 " column matrix (vector)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "It follows from matrix equation (1) that [" }{TEXT 270 1 "K" }{TEXT -1 22 "] is proportional to [" }{TEXT 271 1 "X" }{TEXT -1 5 "], or" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 272 1 "K" }{TEXT -1 1 "]" }{TEXT 602 3 " = " }{XPPEDIT 18 0 "lambda" "6#%'lambda G" }{TEXT -1 2 " [" }{TEXT 273 1 "X" }{TEXT -1 1 "]" }{TEXT 579 79 " \+ \+ (2)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" } {TEXT -1 28 " is some scalar multiplier." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Therefore, equati on (1) may be rewritten in the form" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 274 1 "A" }{TEXT -1 3 "] [" }{TEXT 275 1 "X" } {TEXT -1 1 "]" }{TEXT 603 3 " = " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG " }{TEXT -1 2 " [" }{TEXT 276 1 "X" }{TEXT -1 1 "]" }{TEXT 580 75 " \+ ( 3)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "which is the matrix form of the system of " }{TEXT 277 11 "homogeneous" }{TEXT -1 11 " equations" }}}{EXCHG {PARA 258 "" 0 " " {TEXT 604 1 "(" }{TEXT -1 1 "[" }{TEXT 279 1 "A" }{TEXT -1 1 "]" } {TEXT 605 3 " \226 " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 280 1 "U" }{TEXT -1 1 "]" }{TEXT 606 1 ")" }{TEXT -1 2 " [" }{TEXT 281 1 "X" }{TEXT -1 1 "]" }{TEXT 607 3 " = " }{TEXT -1 1 "[ " }{TEXT 278 1 "0" }{TEXT -1 1 "]" }{TEXT 581 66 " \+ (4)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where [" } {TEXT 284 1 "U" }{TEXT -1 10 "] is the " }{TEXT 282 4 "unit" }{TEXT -1 14 " matrix and [" }{TEXT 283 1 "0" }{TEXT -1 10 "] is the " } {TEXT 756 4 "zero" }{TEXT -1 25 " column matrix (vector)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "T here is a theorem, which states that the " }{TEXT 288 11 "homogeneous " }{TEXT -1 21 " system of equations" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 285 1 "A" }{TEXT -1 3 "] [" }{TEXT 286 1 "X" } {TEXT -1 1 "]" }{TEXT 608 3 " = " }{TEXT -1 1 "[" }{TEXT 287 1 "0" } {TEXT -1 1 "]" }{TEXT 582 82 " \+ (5)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 6 "always" }{TEXT -1 10 " has the " }{TEXT 292 7 "trivial" }{TEXT -1 13 " solution [ " }{TEXT 290 1 "X" }{TEXT -1 1 "]" }{TEXT 609 3 " = " }{TEXT -1 1 "[" }{TEXT 289 1 "0" }{TEXT -1 17 "] and it has a " }{TEXT 293 11 "non-t rivial" }{TEXT -1 11 " solution " }{TEXT 295 4 "only" }{TEXT -1 33 " \+ when the determinant of matrix [" }{TEXT 294 1 "A" }{TEXT -1 6 "] is \+ " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 7 ". If [" }{TEXT 336 1 "X" } {TEXT -1 41 "] is a non-trivial solution, so also is " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 337 1 "X" }{TEXT -1 11 "], where " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 8 " \+ is an " }{TEXT 338 9 "arbitrary" }{TEXT -1 8 " scalar." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "A ccording to this theorem, equation (4) can " }{TEXT 296 4 "only" } {TEXT -1 9 " have a " }{TEXT 297 11 "non-trivial" }{TEXT -1 15 " sol ution when" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 301 4 "Det(" }{TEXT -1 1 "[" }{TEXT 298 1 "A" }{TEXT -1 1 "]" }{TEXT 610 3 " \226 " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 299 1 "U " }{TEXT -1 1 "]" }{TEXT 611 4 ") = " }{TEXT -1 1 "[" }{TEXT 300 1 "0 " }{TEXT -1 1 "]" }{TEXT 583 71 " \+ (6)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "When expanded, this det erminant gives rise to an algebraic polynomial equation of degree " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 6 " in " }{XPPEDIT 18 0 "lambd a" "6#%'lambdaG" }{TEXT -1 13 " of the form" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "lambda^n + alpha[1]*lambda^(n-1) + alpha[2]*lambda ^(n-2)" "6#,()%'lambdaG%\"nG\"\"\"*&&%&alphaG6#F'F')F%,&F&F'F'!\"\"F'F '*&&F*6#\"\"#F')F%,&F&F'F2F.F'F'" }{TEXT 302 9 " + ... + " }{XPPEDIT 18 0 "alpha[n] = 0" "6#/&%&alphaG6#%\"nG\"\"!" }{TEXT 584 28 " \+ (7)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The matrix " }{XPPEDIT 18 0 "lamb da" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 407 1 "U" }{TEXT -1 1 "]" } {TEXT 612 3 " \226 " }{TEXT -1 1 "[" }{TEXT 408 1 "A" }{TEXT -1 18 "] \+ is called the " }{TEXT 410 21 "characteristic matrix" }{TEXT -1 6 " \+ of [" }{TEXT 409 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The determinant \+ " }{TEXT 305 4 "Det(" }{TEXT -1 1 "[" }{TEXT 303 1 "A" }{TEXT -1 1 "] " }{TEXT 613 3 " \226 " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 304 1 "U" }{TEXT -1 1 "]" }{TEXT 614 1 ")" }{TEXT -1 17 " is called the " }{TEXT 306 26 "characteristic determinant" } {TEXT -1 19 " associated with [" }{TEXT 307 1 "A" }{TEXT -1 35 "], an d equation (7) is called the " }{TEXT 309 23 "characteristic equation " }{TEXT -1 6 " of [" }{TEXT 308 1 "A" }{TEXT -1 53 "]. The left-hand side of equation (7) is called the " }{TEXT 317 25 "characteristic p olynomial" }{TEXT -1 6 " of [" }{TEXT 318 1 "A" }{TEXT -1 8 "]. The \+ " }{TEXT 319 23 "characteristic equation" }{TEXT -1 6 " of [" }{TEXT 320 1 "A" }{TEXT -1 7 "] has " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 10 " roots, " }{XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" } {TEXT 310 3 ", " }{XPPEDIT 18 0 "lambda[2]" "6#&%'lambdaG6#\"\"#" } {TEXT 311 9 ", ..., " }{XPPEDIT 18 0 "lambda[n]" "6#&%'lambdaG6#%\"n G" }{TEXT 312 1 "," }{TEXT -1 37 " each of which is called either an \+ " }{TEXT 313 10 "eigenvalue" }{TEXT -1 6 ", a " }{TEXT 314 19 "char acteristic root" }{TEXT -1 25 ", or, in some texts, a " }{TEXT 315 11 "latent root" }{TEXT -1 6 " of [" }{TEXT 316 1 "A" }{TEXT -1 21 "] . The roots may be " }{TEXT 398 4 "real" }{TEXT -1 3 ", " }{TEXT 399 7 "complex" }{TEXT -1 6 ", or " }{TEXT 400 9 "multiples" }{TEXT -1 16 " of each other." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Once the eigenvalues have been de termined, any of them can be substituted into equation (3) to find a c orresponding solution vector [" }{TEXT 335 1 "X" }{TEXT -1 18 "]. Thus , setting " }{XPPEDIT 18 0 "lambda=lambda[i]" "6#/%'lambdaG&F$6#%\"iG " }{TEXT -1 32 " will yield a solution vector " }{XPPEDIT 18 0 "X[i] " "6#&%\"XG6#%\"iG" }{TEXT 339 1 "," }{TEXT -1 113 " which, because o f the aforementioned theorem, will only be determined to within an arb itrary scalar multiplier." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "This vector " }{XPPEDIT 18 0 "X[i ]" "6#&%\"XG6#%\"iG" }{TEXT -1 23 " is called either an " }{TEXT 341 11 "eigenvector" }{TEXT -1 6 ", a " }{TEXT 342 21 "characteristi c vector" }{TEXT -1 9 " or, a " }{TEXT 343 13 "latent vector" } {TEXT -1 6 " of [" }{TEXT 344 1 "A" }{TEXT -1 20 "] corresponding to \+ " }{XPPEDIT 18 0 "lambda[i]" "6#&%'lambdaG6#%\"iG" }{TEXT 340 1 "." } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Eigenvalues of a matrix may be " }{XPPMATH 20 "6#%%zeroG " }{TEXT -1 30 ", whereas an eigenvector may " }{TEXT 415 3 "not" } {TEXT -1 9 " be the " }{TEXT 755 4 "zero" }{TEXT -1 9 " vector." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 615 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 321 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 616 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 322 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := matrix(2, 2 , [1, 2, 3, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%\"AG-%'matrixG6#7$7$\"\"\"\"\"#7$\"\"$\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "and find its characteristic equation, eigenvalues, and eigenvectors." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 325 6 "S tep 1" }{TEXT -1 22 ". Obtain the matrix [" }{TEXT 323 1 "A" }{TEXT -1 1 "]" }{TEXT 617 3 " \226 " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" } {TEXT -1 2 " [" }{TEXT 324 1 "U" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The approp riately sized " }{TEXT 349 4 "unit" }{TEXT -1 38 " matrix correspond ing to this case is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "U := diag(1, 1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"UG-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Therefore," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "l := lambda : `A - lU` := \+ evalm(A - l*U) : A - l*U = matrix(`A - lU`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&%'lambdaGF&%\"UGF&!\"\"-%'matrixG6#7$7$ ,&F&F&F(F*\"\"#7$\"\"$,$F(F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 404 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 406 4 "N.B." } {TEXT -1 15 " The matrix [" }{TEXT 411 1 "A" }{TEXT -1 1 "]" }{TEXT 618 3 " \226 " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 412 1 "U" }{TEXT -1 34 "] may also be obtained using the " } {TEXT 405 7 "charmat" }{TEXT -1 15 " function, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`A - lU` := evalm(-charmat(A, l)) : A - l*U = matrix(`A - lU`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG \"\"\"*&%'lambdaGF&%\"UGF&!\"\"-%'matrixG6#7$7$,&F&F&F(F*\"\"#7$\"\"$, $F(F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 413 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 326 6 "Step 2" }{TEXT -1 24 ". Find the d eterminant " }{TEXT 329 4 "Det(" }{TEXT -1 1 "[" }{TEXT 327 1 "A" } {TEXT -1 1 "]" }{TEXT 619 3 " \226 " }{XPPEDIT 18 0 "lambda" "6#%'lamb daG" }{TEXT -1 2 " [" }{TEXT 328 1 "U" }{TEXT -1 1 "]" }{TEXT 620 1 ") " }{TEXT -1 40 ", or the characteristic polynomial of [" }{TEXT 332 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`d et(A - lU)` := det(`A - lU`) : Det(A - l*U) = `det(A - lU)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#,&%\"AG\"\"\"*&%'lambdaGF)% \"UGF)!\"\",(F+F-*$)F+\"\"#F)F)\"\"'F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 390 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 383 4 "N.B." }{TEXT -1 51 " The characteristic polynomial may be obta ined in " }{TEXT 382 5 "Maple" }{TEXT -1 1 " " }{TEXT 414 8 "directly " }{TEXT -1 12 ", using the " }{TEXT 381 8 "charpoly" }{TEXT -1 36 " f unction. For this particular case," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`charpoly(A)` := charpoly(A, l) : char_poly(A) = `c harpoly(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*char_polyG6#%\"AG ,(%'lambdaG!\"\"*$)F)\"\"#\"\"\"F.\"\"'F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 391 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 330 6 "Step 3" }{TEXT -1 48 ". Form the relevant characteristic equati on of [" }{TEXT 331 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "char_eq(A) := `det(A - lU)` = 0 : char_eq(A) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%'lambdaG!\"\"*$)F%\"\"#\"\"\"F*\" \"'F&\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "or, using " }{TEXT 401 6 "direct" }{TEXT -1 46 " \+ computation of the characteristic polynomial," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "char_eq(A) := `charpoly(A)` = 0 : char_eq(A) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%'lambdaG!\"\"*$)F%\"\"#\"\"\"F *\"\"'F&\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 333 6 "Step 4" }{TEXT -1 40 ". Solve the charact eristic equation of [" }{TEXT 334 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "solution(char_eq(A)) := solve(\{cha r_eq(A)\}, \{l\}) : solution(char_eq(A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#/%'lambdaG\"\"$<#/F%!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Denoting the root s by " }{XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "lambda[2]" "6#&%'lambdaG6#\"\"#" }{TEXT -1 65 " and extracting either of them from the solution sequence yields " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "l[`1_n`] := lambda[1] \+ : l[`2_n`] := lambda[2] : l[`1_v`] := subs(solution(char_eq(A))[1] , lambda) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "l[`2_v`] := \+ subs(solution(char_eq(A))[2], lambda) : l[`1_n`] = l[`1_v`] ; l[`2 _n`] = l[`2_v`] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\" \"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"#!\"#" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The roots " }{XPPEDIT 18 0 "lambda[1]=3" "6#/&%'lambdaG6#\"\" \"\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "lambda[2]=-2" "6#/&%'la mbdaG6#\"\"#,$F'!\"\"" }{TEXT -1 50 " are the eigenvalues or characte ristic roots of [" }{TEXT 345 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 388 5 "* * * " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 387 4 "N.B." }{TEXT -1 37 " The eigenvalues may be obtained in \+ " }{TEXT 385 5 "Maple" }{TEXT -1 1 " " }{TEXT 386 8 "directly" }{TEXT -1 12 ", using the " }{TEXT 384 9 "eigenvals" }{TEXT -1 36 " function. For this particular case," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$\"\"$!\"#" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Denoting the characteristic roots by " }{XPPEDIT 18 0 "lambda[ 1]" "6#&%'lambdaG6#\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "lambd a[2]" "6#&%'lambdaG6#\"\"#" }{TEXT -1 52 " and extracting each from t he above sequence yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "l[`1_v`] := charroots(A)[1] : l[`2_v`] := charroots(A)[2] : r oot1(A) := l[`1_v`] : root2(A) := l[`2_v`] :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "l[`1_n`] = l[`1_v`] ; l[`2_n`] = l[`2_v`] ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"#!\"#" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Alternativ ely, the inert " }{TEXT 396 9 "Eigenvals" }{TEXT -1 53 " function may \+ be used, but it must be evaluated with " }{TEXT 397 5 "evalf" }{TEXT -1 16 ". For this case," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 " charroots(A) := evalf(Eigenvals(A)) : char_roots(A) = charroots(A) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG-%'vectorG6# 7$$\"+++++I!\"*$!+++++?F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "The returned output is an array s tructure, from which the characteristic roots may be extracted using s ubscript notation, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " charroots(A)[1] ; charroots(A)[2] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++I!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+++++?!\"*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 389 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 346 6 "Step 5" }{TEXT -1 32 ". Determine the eig envector of [" }{TEXT 348 1 "A" }{TEXT -1 20 "] corresponding to " } {XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" }{TEXT 347 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 754 4 "zero" }{TEXT -1 61 " column matrix (vector ) appropriately sized for this case is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`0` := matrix(2,1, [0, 0]) : `0` = matrix(`0`) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"0G-%'matrixG6#7$7#\"\"!F)" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(a) The eigenvector " }{XPPEDIT 18 0 "X[lambda1]" "6#&%\"XG6#% (lambda1G" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "x[1][l1_n] := x[1][l[`1_n`]] : x[2][l1_n] := x[2][l[`1_n`]] : X[l1_n] := X[lambda[1]] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "X[l1_sv] := matrix(2, 1, [x[1][l1_n], x[2][l1_n]]) : X[l1_n] = m atrix(X[l1_sv]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambd aG6#\"\"\"-%'matrixG6#7$7#&&%\"xGF)F&7#&&F26#\"\"#F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(b) The matrix [" }{TEXT 350 1 "A" }{TEXT -1 1 "]" }{TEXT 621 3 " \226 " } {XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" }{TEXT -1 2 " [" } {TEXT 351 1 "U" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "`f(A - l1U)` := subs(lambda=l[`1_n`], matrix(`A - lU` )) : A - l[`1_n`]*U = matrix(`f(A - l1U)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambdaG6#F&F&%\"UGF&!\"\"-%'matrixG6 #7$7$,&F&F&F(F,\"\"#7$\"\"$,$F(F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "or, upon substitution of t he numerical value of " }{XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\" \"\"" }{TEXT 367 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "` v(A - l1U)` := subs(l[`1_n`]=l[`1_v`], matrix(`f(A - l1U)`)) : A - l [`1_n`]*U = matrix(`v(A - l1U)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,&%\"AG\"\"\"*&&%'lambdaG6#F&F&%\"UGF&!\"\"-%'matrixG6#7$7$!\"#\"\"#7 $\"\"$!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "(c) The matrix equation [" }{TEXT 352 1 "A" } {TEXT -1 1 "]" }{TEXT 622 3 " \226 " }{XPPEDIT 18 0 "lambda[1]" "6#&%' lambdaG6#\"\"\"" }{TEXT -1 2 " [" }{TEXT 353 1 "U" }{TEXT -1 1 "]" } {TEXT 623 3 " = " }{TEXT -1 1 "[" }{TEXT 354 1 "0" }{TEXT -1 5 "] is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "matrix(`v(A - l1U)`) * \+ matrix(X[l1_sv]) = matrix(`0`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &-%'matrixG6#7$7$!\"#\"\"#7$\"\"$!\"$\"\"\"-F&6#7$7#&&%\"xG6#F/6#&%'la mbdaGF77#&&F66#F+F8F/-F&6#7$7#\"\"!FB" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "(d) The correspon ding system of homogeneous equations is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "s_l1 := evalm(matrix(`v(A - l1U)`) &* matrix(X[l1_sv] )) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Eq_1_l1 := s_l1[1,1 ] = 0 : Eq_1_l1 ; Eq_2_l1 := s_l1[2,1] = 0 : Eq_2_l1 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&&%\"xG6#\"\"\"6#&%'lambdaGF(!\"#*&\"\"#F )&&F'6#F/F*F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&&%\"xG6#\" \"\"6#&%'lambdaGF(\"\"$*&F-F)&&F'6#\"\"#F*F)!\"\"\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "( e) The solution of the system is obtained as follows:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "A ssign an arbitrary non-" }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 12 " valu e to " }{XPPEDIT 18 0 "x[1][lambda[1]]" "6#&&%\"xG6#\"\"\"6#&%'lambda G6#F'" }{TEXT 372 1 "," }{TEXT -1 6 " e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x[1][l1_v] := 1 : x[1][l1_n] = x[1][l1_v] ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%\"xG6#\"\"\"6#&%'lambdaGF'F(" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Substitute this value in, say, the second equation and solve it for the other unknown, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "x[2][l1_v] := solve(subs(x[1][l1_n]=1, Eq_2_l1)) : x[2][l1_n] \+ = x[2][l1_v] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%\"xG6#\"\"#6#&%' lambdaG6#\"\"\"F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "(f) Substitute the roots " }{XPPEDIT 18 0 "x[1][lambda[1]]" "6#&&%\"xG6#\"\"\"6#&%'lambdaG6#F'" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "x[2][lambda[1]]" "6#&&%\"xG6#\"\"#6#&%'lambdaG 6#\"\"\"" }{TEXT -1 50 " in the symbolic expression for the eigenvect or " }{XPPEDIT 18 0 "X[lambda[1]]" "6#&%\"XG6#&%'lambdaG6#\"\"\"" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "X[l1_nv] : = subs(x[1][l1_n]=x[1][l1_v], x[2][l1_n]=x[2][l1_v], matrix(X[l1_sv])) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "X[l1_n] = matrix(X[l1 _nv]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"\" -%'matrixG6#7$7#F*F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "(g) Check the resultant of the mat rix operation [" }{TEXT 355 1 "A" }{TEXT -1 1 "]" }{TEXT 624 3 " \226 " }{XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" }{TEXT -1 2 " [" }{TEXT 356 1 "U" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A - l[`1_n`] * U = matrix(`v(A - l1U)`) * matrix(X[l1 _nv]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambdaG6 #F&F&%\"UGF&!\"\"*&-%'matrixG6#7$7$!\"#\"\"#7$\"\"$!\"$F&-F/6#7$7#F&F; F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A - l[`1_ n`] * U = evalm(matrix(`v(A - l1U)`) &* matrix(X[l1_nv])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambdaG6#F&F&%\"UGF&!\"\" -%'matrixG6#7$7#\"\"!F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 258 "" 0 "" {TEXT 359 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 360 4 "N.B." }{TEXT -1 28 " For any arbitrary scalar " }{XPPEDIT 18 0 "mu[1]" "6#&%#muG6 #\"\"\"" }{TEXT 357 1 "," }{TEXT -1 22 " the product vector " } {XPPEDIT 18 0 "mu[1]*X[lambda[1]]" "6#*&&%#muG6#\"\"\"F'&%\"XG6#&%'lam bdaG6#F'F'" }{TEXT -1 30 " is also the eigenvector of [" }{TEXT 358 1 "A" }{TEXT -1 7 "], i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "A - l[`1_n`] * U = mu[1] * X[l1_n] ; A - l[`1_n`] * U = mu[1] * \+ matrix(X[l1_sv]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*& &%'lambdaG6#F&F&%\"UGF&!\"\"*&&%#muGF*F&&%\"XG6#F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambdaG6#F&F&%\"UGF&!\"\"*&&%#muG F*F&-%'matrixG6#7$7#&&%\"xGF*6#F(7#&&F76#\"\"#F8F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "since the resultant of the matrix equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A - l[`1_n`] * U = matrix(`v(A - l1U)`) * matrix(eval m(mu[1] * X[l1_nv])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\" \"*&&%'lambdaG6#F&F&%\"UGF&!\"\"*&-%'matrixG6#7$7$!\"#\"\"#7$\"\"$!\"$ F&-F/6#7$7#&%#muGF*F;F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "A - l[`1_n`] * U = evalm(matrix(`v(A - l1U)`) &* matr ix(evalm(mu[1]*X[l1_nv]))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\" AG\"\"\"*&&%'lambdaG6#F&F&%\"UGF&!\"\"-%'matrixG6#7$7#\"\"!F1" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Notice, therefore, that the eigenvector of a matrix is " } {TEXT 736 3 "not" }{TEXT -1 8 " unique." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 361 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 362 6 "Step 6" }{TEXT -1 32 ". Determine the eigenvector of [" }{TEXT 363 1 "A" }{TEXT -1 20 "] corresponding to " }{XPPEDIT 18 0 "lambda[2 ]" "6#&%'lambdaG6#\"\"#" }{TEXT 364 2 " :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(a) The eigenvect or " }{XPPEDIT 18 0 "X[lambda[2]]" "6#&%\"XG6#&%'lambdaG6#\"\"#" } {TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "x[1][l2 _n] := x[1][l[`2_n`]] : x[2][l2_n] := x[2][l[`2_n`]] : X[l2_n] := \+ X[lambda[2]] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "X[l2_sv] \+ := matrix(2, 1, [x[1][l2_n], x[2][l2_n]]) : X[l2_n] = matrix(X[l2_sv ]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"#-%'m atrixG6#7$7#&&%\"xG6#\"\"\"F&7#&&F2F)F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(b) The matrix [ " }{TEXT 365 1 "A" }{TEXT -1 1 "]" }{TEXT 625 3 " \226 " }{XPPEDIT 18 0 "lambda[2]" "6#&%'lambdaG6#\"\"#" }{TEXT -1 2 " [" }{TEXT 366 1 "U" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "`f(A \+ - l2U)` := subs(lambda=l[`2_n`], matrix(`A - lU`)) : A - l[`2_n`]*U \+ = matrix(`f(A - l2U)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\" \"\"*&&%'lambdaG6#\"\"#F&%\"UGF&!\"\"-%'matrixG6#7$7$,&F&F&F(F-F+7$\" \"$,$F(F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 49 "or, upon substitution of the numerical value of \+ " }{XPPEDIT 18 0 "lambda[2]" "6#&%'lambdaG6#\"\"#" }{TEXT 368 1 "," }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "`v(A - l2U)` := subs(l[`2_ n`]=l[`2_v`], matrix(`f(A - l2U)`)) : A - l[`2_n`]*U = matrix(`v(A - l2U)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambda G6#\"\"#F&%\"UGF&!\"\"-%'matrixG6#7$7$\"\"$F+F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "(c) The matr ix equation [" }{TEXT 369 1 "A" }{TEXT -1 1 "]" }{TEXT 626 3 " \226 \+ " }{XPPEDIT 18 0 "lambda[2]" "6#&%'lambdaG6#\"\"#" }{TEXT -1 2 " [" } {TEXT 370 1 "U" }{TEXT -1 1 "]" }{TEXT 627 3 " = " }{TEXT -1 1 "[" } {TEXT 371 1 "0" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "matrix(`v(A - l2U)`) * matrix(X[l2_sv]) = matrix(`0`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7$7$\"\"$\"\"#F) \"\"\"-F&6#7$7#&&%\"xG6#F,6#&%'lambdaG6#F+7#&&F3F8F5F,-F&6#7$7#\"\"!F? " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "(d) The corresponding system of (identical) homogeneous e quations is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "s_l2 := eval m(matrix(`v(A - l2U)`) &* matrix(X[l2_sv])) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Eq_1_l2 := s_l2[1,1] = 0 : Eq_1_l2 ; Eq_2_l2 := s_l2[2,1] = 0 : Eq_2_l2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,& &&%\"xG6#\"\"\"6#&%'lambdaG6#\"\"#\"\"$*&F.F)&&F'F-F*F)F)\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&&%\"xG6#\"\"\"6#&%'lambdaG6#\"\"# \"\"$*&F.F)&&F'F-F*F)F)\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "(e) The solution of the system \+ is obtained as follows:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Assign an arbitrary non-" } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 11 " value to " }{TEXT 373 1 " " } {XPPEDIT 18 0 "x[1][lambda[2]]" "6#&&%\"xG6#\"\"\"6#&%'lambdaG6#\"\"# " }{TEXT 771 1 "," }{TEXT -1 6 " e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x[1][l2_v] := 1 : x[1][l2_n] = x[1][l2_v] ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%\"xG6#\"\"\"6#&%'lambdaG6#\"\"#F( " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Substitute this value in, say, the second equation and so lve it for the other unknown, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "x[2][l2_v] := solve(subs(x[1][l2_n]=1, Eq_2_l2)) : \+ x[2][l2_n] = x[2][l2_v] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%\"xG6 #\"\"#6#&%'lambdaGF'#!\"$F(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(f) Substitute the roots " } {TEXT 772 1 " " }{XPPEDIT 18 0 "x[1][lambda[2]]" "6#&&%\"xG6#\"\"\"6#& %'lambdaG6#\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "x[2][lambda[2] ]" "6#&&%\"xG6#\"\"#6#&%'lambdaG6#F'" }{TEXT -1 50 " in the symbolic \+ expression for the eigenvector " }{XPPEDIT 18 0 "X[lambda[2]]" "6#&% \"XG6#&%'lambdaG6#\"\"#" }{TEXT 451 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "X[l2_nv] := subs(x[1][l2_n]=x[1][l2_v], x[2][l2_n]=x[ 2][l2_v], matrix(X[l2_sv])) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "X[l2_n] = matrix(X[l2_nv]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"XG6#&%'lambdaG6#\"\"#-%'matrixG6#7$7#\"\"\"7##!\"$F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "( g) Check the resultant of the matrix operation [" }{TEXT 374 1 "A" } {TEXT -1 1 "]" }{TEXT 628 3 " \226 " }{XPPEDIT 18 0 "lambda[2]" "6#&%' lambdaG6#\"\"#" }{TEXT -1 2 " [" }{TEXT 375 1 "U" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A - l[`2_n`] * U = matrix(`v (A - l2U)`) * matrix(X[l2_nv]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, &%\"AG\"\"\"*&&%'lambdaG6#\"\"#F&%\"UGF&!\"\"*&-%'matrixG6#7$7$\"\"$F+ F3F&-F06#7$7#F&7##!\"$F+F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A - l[`2_n`] * U = evalm(matrix(`v(A - l2U)`) &* matr ix(X[l2_nv])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%' lambdaG6#\"\"#F&%\"UGF&!\"\"-%'matrixG6#7$7#\"\"!F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 376 5 "* * * " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 379 4 "N.B." }{TEXT -1 28 " For any arbitrary scalar " } {XPPEDIT 18 0 "mu[2]" "6#&%#muG6#\"\"#" }{TEXT 377 1 "," }{TEXT -1 22 " the product vector " }{XPPEDIT 18 0 "mu[2]*X[lambda[2]]" "6#*&&%#m uG6#\"\"#\"\"\"&%\"XG6#&%'lambdaG6#F'F(" }{TEXT -1 30 " is also the e igenvector of [" }{TEXT 378 1 "A" }{TEXT -1 7 "], i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "A - l[`2_n`] * U = mu[2] * X[l2_n] \+ ; A - l[`2_n`] * U = mu[2] * matrix(X[l2_sv]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambdaG6#\"\"#F&%\"UGF&!\"\"*&&%#muG F*F&&%\"XG6#F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&% 'lambdaG6#\"\"#F&%\"UGF&!\"\"*&&%#muGF*F&-%'matrixG6#7$7#&&%\"xG6#F&6# F(7#&&F8F*F:F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "since the resultant of the matrix equatio n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A - l[`2_n`] * U = mat rix(`v(A - l2U)`) * matrix(evalm(mu[2] * X[l2_nv])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambdaG6#\"\"#F&%\"UGF&!\"\"*&- %'matrixG6#7$7$\"\"$F+F3F&-F06#7$7#&%#muGF*7#,$F9#!\"$F+F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "A - l[`2_n`] * U = eval m(matrix(`v(A - l2U)`) &* matrix(evalm(mu[2]*X[l2_nv]))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"*&&%'lambdaG6#\"\"#F&%\"UGF&! \"\"-%'matrixG6#7$7#\"\"!F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 258 "" 0 "" {TEXT 380 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 395 4 "N.B." }{TEXT -1 38 " The eigenvectors may be obtained in " }{TEXT 393 5 "Maple" } {TEXT -1 1 " " }{TEXT 394 8 "directly" }{TEXT -1 12 ", using the " } {TEXT 392 10 "eigenvects" }{TEXT -1 39 " function. This function compu tes the " }{TEXT 457 12 "eigenvalues " }{TEXT -1 1 " " }{TEXT 459 3 " and" }{TEXT -1 2 " " }{TEXT 458 12 "eigenvectors" }{TEXT -1 41 " of \+ a matrix. The result returned is a " }{TEXT 460 17 "sequence of lists " }{TEXT -1 19 ". For the matrix [" }{TEXT 675 1 "A" }{TEXT -1 22 "] \+ analysed above, the " }{TEXT 674 10 "eigenvects" }{TEXT -1 16 " functi on yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`roots&vectors (A)` := eigenvects(A) : roots_and_vectors(A) = `roots&vectors(A)` ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6#%\"AG6$7% \"\"$\"\"\"<#-%'vectorG6#7$F+F+7%!\"#F+<#-F.6#7$F+#!\"$\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "However, the " }{TEXT 687 10 "eigenvects" }{TEXT -1 181 " funct ion returns an unordered sequence of its elements, i.e. lists may appe ar in the above or reverse order. This implies that the first eigenval ue in the sequence returned by the " }{TEXT 689 10 "eigenvects" } {TEXT -1 86 " function may not be the same as the first eigenvalue in \+ the sequence returned by the " }{TEXT 688 9 "eigenvals" }{TEXT -1 69 " function, which returns eigenvalues in a unique, reproducible order. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 272 "On the other hand, it is essential that every eigenvalue be used together with the eigenvector corresponding to it. Therefore, it is necessary to develop a procedure that will extract pairs of lis ts in a unique way. This is ensured by the selection procedure that fo llows." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "For the case under consideration, two different list s of " }{TEXT 690 11 "eigenvalues" }{TEXT -1 7 " and " }{TEXT 691 12 "eigenvectors" }{TEXT -1 66 " are extracted from the above solutio n in the desired order, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "e[1] := `roots&vectors(A)`[1][1] : e[2] := `roots&vectors(A)`[ 2][1] :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 676 1 "\225" }{TEXT -1 39 " list containing the first eige nvalue:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "if e[1] = root 1(A) then l := root1(A) : list1 := `roots&vectors(A)`[1] else l \+ := e[2] : list1 := `roots&vectors(A)`[2] fi : list[1](roots_and_v ectors(A)) = list1 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%%listG6#\" \"\"6#-%2roots_and_vectorsG6#%\"AG7%\"\"$F(<#-%'vectorG6#7$F(F(" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 677 1 "\225" }{TEXT -1 40 " list containing the second eigenvalue:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "if e[2] = root2(A) then l := root2(A) : list2 := `roots&vectors(A)`[2] else l := e[1] : list2 := `roots&vectors(A)`[1] fi : list[2](roots_and_vectors(A)) = list2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%%listG6#\"\"#6#-%2ro ots_and_vectorsG6#%\"AG7%!\"#\"\"\"<#-%'vectorG6#7$F0#!\"$F(" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "In either list, the " }{TEXT 678 5 "first" }{TEXT -1 15 " numbe r is an " }{TEXT 679 10 "eigenvalue" }{TEXT -1 10 " and the " } {TEXT 680 6 "second" }{TEXT -1 15 " value is its " }{TEXT 681 12 "mul tiplicity" }{TEXT -1 219 ", indicating how many times this eigenvalue appears in the solution to the characteristic equation of the matrix. This may be verified by inspection of the characteristic polynomial i n its factored form. For this case," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "`charpoly(A)` := factor(charpoly(A, lambda)) : char _poly(A) = `charpoly(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*char _polyG6#%\"AG*&,&%'lambdaG\"\"\"\"\"#F+F+,&F*F+\"\"$!\"\"F+" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "It is evident from the above that multiplicity of either eigenv alue is " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 29 ", or either eige nvalue is a " }{TEXT 487 6 "simple" }{TEXT -1 42 " root of the chara cteristic equation of [" }{TEXT 682 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 215 " The last object in the solution list is a set containing an eigenvecto r (or, in the case of matrices of a higher order, several eigenvectors separated with the comma) corresponding to the eigenvalue of a given \+ list." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Extracting the eigenvalue and its corresponding eigenv ector from the first list gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "l1 := list1[1] : charvector[1] := list1[3][1] : v1 := char vector[1] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "lambda[1] = \+ l1 ; char_vector[lambda[1]](A) = eval(charvector[1]) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%,char_vectorG6#&%'lambdaG6#\"\"\"6#%\"AG-%'vectorG6 #7$F+F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "or, as a column matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "X[l1] := convert(eval(charvector[1]), matri x) : X[lambda[1]] = matrix(X[l1]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"\"-%'matrixG6#7$7#F*F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Extractin g the eigenvalue and its corresponding eigenvector from the second lis t gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "l2 := list2[1] \+ : charvector[2] := list2[3][1] : v2 := charvector[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "lambda[2] = l2 ; char_vector[lamb da[2]](A) = eval(charvector[2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%'lambdaG6#\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%,char_vec torG6#&%'lambdaG6#\"\"#6#%\"AG-%'vectorG6#7$\"\"\"#!\"$F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "o r, as a column matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "X[l2] := convert(eval(charvector[2]), matrix) : X[lambda[2]] = matrix(X[l2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lamb daG6#\"\"#-%'matrixG6#7$7#\"\"\"7##!\"$F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Both eigenvector s are equal to the eigenvectors obtained earlier \"manually\" and eith er of them is associated with a proper eigenvalue." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 686 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "mu[1]*X[lambda[1]]" "6#*&&%#muG6# \"\"\"F'&%\"XG6#&%'lambdaG6#F'F'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "mu[2]*X[lambda[2]]" "6#*&&%#muG6#\"\"#\"\"\"&%\"XG6#&%'lambdaG6#F'F (" }{TEXT -1 28 " are also eigenvectors of [" }{TEXT 683 1 "A" } {TEXT -1 31 "], this fact is often used to " }{TEXT 684 9 "normalize " }{TEXT -1 51 " eigenvectors of a matrix by setting the scalars " } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 44 " to be equal to the the re ciprocal of the " }{TEXT 692 9 "Frobenius" }{TEXT -1 2 " " }{TEXT 693 4 "norm" }{TEXT -1 26 " of the pertinent vector." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Th e " }{TEXT 694 9 "Frobenius" }{TEXT -1 2 " " }{TEXT 695 4 "norm" } {TEXT -1 15 " of a vector [" }{TEXT 696 1 "X" }{TEXT -1 10 "] having \+ " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 12 " elements " }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 137 " is defined to be the sq uare root of the sum of the squares of the magnitudes of each element \+ of the vector. In other words, it is the " }{TEXT 697 6 "length" } {TEXT -1 49 " (or magnitude) of the vector, which is given by" }}} {EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(X) = sqrt(Sum((abs(x[i]))^ 2, i=1..n))" "6#/-%$absG6#%\"XG-%%sqrtG6#-%$SumG6$*$-F%6#&%\"xG6#%\"iG \"\"#/F4;\"\"\"%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "For the case under analysis, the \+ " }{TEXT 698 9 "Frobenius" }{TEXT -1 34 " norm or length of eigenvect ors " }{XPPEDIT 18 0 "X[lambda[1]]" "6#&%\"XG6#&%'lambdaG6#\"\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "X[lambda[2]]" "6#&%\"XG6#&%'lambd aG6#\"\"#" }{TEXT -1 29 " may be computed as follows." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 685 1 " \225" }{TEXT -1 56 " for eigenvector corresponding to the first eigen value:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`abs(v1)` := sqrt (v1[1]^2 + v1[2]^2) : abs(char_vector[lambda[1]](A)) = `abs(v1)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#-&%,char_vectorG6#&%'lambd aG6#\"\"\"6#%\"AG*$-%%sqrtG6#\"\"#F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "or, directly by using t he function " }{TEXT 699 4 "norm" }{TEXT -1 29 " together with the nor m name " }{TEXT 700 9 "frobenius" }{TEXT -1 6 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`norm(v1)` := norm(v1, frobenius) \+ : abs(char_vector[lambda[1]](A)) = `norm(v1)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#-&%,char_vectorG6#&%'lambdaG6#\"\"\"6#%\"AG*$ -%%sqrtG6#\"\"#F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 701 1 "\225" }{TEXT -1 57 " for eigenvector cor responding to the second eigenvalue:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`abs(v2)` := sqrt(v2[1]^2 + v2[2]^2) : abs(char_vec tor[lambda[2]](A)) = `abs(v2)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$absG6#-&%,char_vectorG6#&%'lambdaG6#\"\"#6#%\"AG,$*$-%%sqrtG6#\"#8\" \"\"#F7F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 13 "or, directly," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`norm(v2)` := norm(v2, frobenius) : abs(char_vector [lambda[2]](A)) = `norm(v2)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ absG6#-&%,char_vectorG6#&%'lambdaG6#\"\"#6#%\"AG,$*$-%%sqrtG6#\"#8\"\" \"#F7F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Consequently, the scalars " }{XPPEDIT 18 0 "mu" "6# %#muG" }{TEXT -1 30 " assume the following values:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "mu[1] := 1/`abs(v1)` : mu[2] := 1/`abs( v2)` : 'mu[1]' = mu[1] ; 'mu[2]' = mu[2] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#muG6#\"\"\",$*$-%%sqrtG6#\"\"#F'#F'F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#muG6#\"\"#,$*$-%%sqrtG6#\"#8\"\"\"#F'F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The normalized characteristic vectors of [" }{TEXT 702 1 "A" } {TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "`v1[N ]` := evalm(mu[1] * v1) : `v2[N]` := evalm(mu[2] * v2) : mu[1] := \+ 'mu[1]' : mu[2] := 'mu[2]' :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "char_vector[lambda[1]][N](A) = op(`v1[N]`) ; char_vector[lam bda[2]][N](A) = op(`v2[N]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&% ,char_vectorG6#&%'lambdaG6#\"\"\"6#%\"NG6#%\"AG-%'vectorG6#7$,$*$-%%sq rtG6#\"\"#F,#F,F:F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%,char_vect orG6#&%'lambdaG6#\"\"#6#%\"NG6#%\"AG-%'vectorG6#7$,$*$-%%sqrtG6#\"#8\" \"\"#F,F:,$F6#!\"$F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Alternatively, normalization of ve ctors may be performed in " }{TEXT 703 5 "Maple" }{TEXT -1 1 " " } {TEXT 705 8 "directly" }{TEXT -1 12 ", using the " }{TEXT 704 9 "norma lize" }{TEXT -1 44 " function. In this particular case, it gives" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`v1[N]` := normalize(v1) : \+ `v2[N]` := normalize(v2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "char_vector[lambda[1]][N](A) = op(`v1[N]`) ; char_vector[lambda[ 2]][N](A) = op(`v2[N]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%,cha r_vectorG6#&%'lambdaG6#\"\"\"6#%\"NG6#%\"AG-%'vectorG6#7$,$*$-%%sqrtG6 #\"\"#F,#F,F:F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%,char_vectorG6 #&%'lambdaG6#\"\"#6#%\"NG6#%\"AG-%'vectorG6#7$,$*&-%%sqrtG6#\"#8\"\"\" -F86#\"\"%F;#F;F:,$F6#!\"$\"#E" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Note that the magnitude of no rmalized vectors is " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 56 ". Veri fication of this fact for the analysed case gives" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 105 "`abs(v1[N])` := sqrt(`v1[N]`[1]^2 + `v1[N]` [2]^2) : `abs(v2[N])` := sqrt(`v2[N]`[1]^2 + `v2[N]`[2]^2) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "abs(char_vector[lambda[1]][ N](A)) = `abs(v1[N])` ; abs(char_vector[lambda[2]][N](A)) = `abs(v2[ N])` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#-&&%,char_vectorG6 #&%'lambdaG6#\"\"\"6#%\"NG6#%\"AGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$absG6#-&&%,char_vectorG6#&%'lambdaG6#\"\"#6#%\"NG6#%\"AG\"\"\"" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 416 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 707 4 "N.B." }{TEXT -1 44 " If one of the eigen values of a matrix is " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 23 ", its determinant is " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 29 " and the ma trix is singular." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 710 1 " (" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 709 3 " \327 " }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 711 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 708 1 " A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A := matrix(3, 3, [1, 2, 5, 3, 1, 5, -5, 0, -5]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\" #\"\"&7%\"\"$F*F,7%!\"&\"\"!F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenvalues of [" } {TEXT 712 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = char roots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6% \"\"!,&#!\"$\"\"#\"\"\"*&^##F.F-F.-%%sqrtG6#\"#JF.F.,&F+F.*&^##!\"\"F- F.F2F.F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 24 "(b) The determinant of [" }{TEXT 713 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := d et(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$D etG6#%\"AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 714 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 727 4 "N.B." }{TEXT -1 6 " If " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 32 " is an eigenv alue of a matrix [" }{TEXT 726 1 "A" }{TEXT -1 26 "], then an eigenval ue of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 " [" }{TEXT 728 1 "A " }{TEXT -1 7 "] is " }{XPPEDIT 18 0 "k*lambda" "6#*&%\"kG\"\"\"%'la mbdaGF%" }{TEXT 735 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 731 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 730 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 732 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 729 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A := matrix(3, 3, [-2, 2, 0, 2, 1, 0, -2, -1, -1]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matri xG6#7%7%!\"#\"\"#\"\"!7%F+\"\"\"F,7%F*!\"\"F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenv alues of [" }{TEXT 733 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6# %\"AG6%!\"\"\"\"#!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(b) The eigenvalues of " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 " [" }{TEXT 734 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "charroots(kA ) := eigenvals(evalm(k*A)) : char_roots(k*A) = charroots(kA) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#*&%\"kG\"\"\"%\"AGF)6 %,$F(!\"\",$F(\"\"#,$F(!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 258 "" 0 "" {TEXT 725 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 449 4 "N.B." }{TEXT -1 6 " If " }{XPPEDIT 18 0 "X[1]" "6#&%\"XG6#\"\"\"" }{TEXT 452 1 ", " }{TEXT -1 2 " " }{XPPEDIT 18 0 "X[2]" "6#&%\"XG6#\"\"#" }{TEXT 453 1 "," }{TEXT -1 1 " " }{TEXT 450 5 "..., " }{XPPEDIT 18 0 "X[n]" "6#&% \"XG6#%\"nG" }{TEXT -1 36 " are all eigenvectors of a matrix [" } {TEXT 455 1 "A" }{TEXT -1 23 "] corresponding to the " }{TEXT 461 4 "s ame" }{TEXT -1 13 " eigenvalue " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG " }{TEXT 454 1 "," }{TEXT -1 15 " then any non-" }{XPPMATH 20 "6#%%ze roG" }{TEXT -1 65 " linear combination of these vectors is also an ei genvector of [" }{TEXT 456 1 "A" }{TEXT -1 20 "] corresponding to " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 31 ". This implies tha t the vector" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "X[lambda] = m u[1]*X[1][lambda] + mu[2]*X[2][lambda]" "6#/&%\"XG6#%'lambdaG,&*&&%#mu G6#\"\"\"F-&&F%6#F-6#F'F-F-*&&F+6#\"\"#F-&&F%6#F56#F'F-F-" }{TEXT 482 9 " + ... + " }{XPPEDIT 18 0 "mu[n]*X[n][lambda]" "6#*&&%#muG6#%\"nG\" \"\"&&%\"XG6#F'6#%'lambdaGF(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "is a solution vector to the equ ation" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 473 2 "([" }{TEXT 474 1 "A" } {TEXT 475 4 "] \226 " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 480 2 " [" }{TEXT 476 1 "U" }{TEXT 477 3 "]) " }{XPPEDIT 18 0 "X[lambd a]" "6#&%\"XG6#%'lambdaG" }{TEXT 481 4 " = [" }{TEXT 478 1 "0" }{TEXT 479 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 30 "with the selected eigenvalue " }{XPPEDIT 18 0 "la mbda" "6#%'lambdaG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Coefficients " } {XPPEDIT 18 0 "mu[1]" "6#&%#muG6#\"\"\"" }{TEXT 639 3 ", " }{XPPEDIT 18 0 "mu[2]" "6#&%#muG6#\"\"#" }{TEXT 640 7 ", ..., " }{XPPEDIT 18 0 " mu[n]" "6#&%#muG6#%\"nG" }{TEXT -1 24 " are arbitrary scalars." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 629 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 463 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 630 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 462 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(3, 3 , [-2, 2, 3, 2, 1, 6, 3, 6, 6]) : A = matrix(A) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%!\"#\"\"#\"\"$7%F+\"\"\"\"\"'7% F,F/F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenvalues of [" }{TEXT 483 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6%\"#6!\"$F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "from wh ich two distinct roots are extracted, viz." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "root1(A) := charroots(A)[1] : root2(A) := charroo ts(A)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1]( A) = root1(A) ; char_root[2](A) = root2(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\"AG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"#6#%\"AG!\"$" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The eigenv alue " }{XPPMATH 20 "6#!\"$" }{TEXT -1 8 " is a " }{TEXT 488 6 "dou ble" }{TEXT -1 104 " root (root of multiplicity two), as clearly seen from inspection of the characteristic polynomial of [" }{TEXT 486 1 " A" }{TEXT -1 22 "] in its factored form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "`charpoly(A)` := factor(charpoly(A, lambda)) : char _poly(A) = `charpoly(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*char _polyG6#%\"AG*&,&%'lambdaG\"\"\"\"#6!\"\"F+),&F*F+\"\"$F+\"\"#F+" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(b) The" }{TEXT 464 1 " " }{TEXT -1 62 "sequence of lists contai ning eigenvalues and eigenvectors of [" }{TEXT 465 1 "A" }{TEXT -1 4 " ] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`roots&vectors(A)` \+ := eigenvects(A) : roots_and_vectors(A) = `roots&vectors(A)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6#%\"AG6$7%\"#6 \"\"\"<#-%'vectorG6#7%#F+\"\"#F+#\"\"$F27%!\"$F2<$-F.6#7%F6\"\"!F+-F.6 #7%!\"#F+F;" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "It can be seen from the above sequence that onl y " }{TEXT 484 3 "one" }{TEXT -1 13 " eigenvalue " }{TEXT 631 2 "( " }{XPPMATH 20 "6#!\"$" }{TEXT 632 2 " )" }{TEXT -1 21 " is associated \+ with " }{TEXT 485 3 "two" }{TEXT -1 152 " eigenvectors. This eigenvalu e together with its multiplicity and both corresponding eigenvectors a re contained in one of the two lists of the sequence." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "S ince only " }{XPPEDIT 18 0 "eigenvalue =-3" "6#/%+eigenvalueG,$\"\"$! \"\"" }{TEXT -1 247 " and its corresponding eigenvectors are of inter est in this case, it is necessary to develop a selection procedure, wh ich will extract these items from the sequence irrespective of the act ual order of the lists in it. This is performed under (c)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "( c) The eigenvalue involved and the sequence containing eigenvectors co rresponding to it are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "e[ 1] := `roots&vectors(A)`[1][1] : e[2] := `roots&vectors(A)`[2][1] : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "if e[1] = root1(A) t hen l := root2(A) : vectors(A) := op(`roots&vectors(A)`[2][3]) els e l := e[1] : vectors(A) := op(`roots&vectors(A)`[1][3]) fi :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "l := l : lambda = l ; ch ar_vectors[lambda](A) = vectors(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%'lambdaG!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%-char_vectorsG 6#%'lambdaG6#%\"AG6$-%'vectorG6#7%!\"#\"\"\"\"\"!-F-6#7%!\"$F2F1" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(d) Assigning the names " }{XPPEDIT 18 0 "X[1][lambda]" "6#&&% \"XG6#\"\"\"6#%'lambdaG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "X[2][la mbda]" "6#&&%\"XG6#\"\"#6#%'lambdaG" }{TEXT -1 102 " to the eigenvect ors of the above sequence, extracting them, and converting into column matrices give" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "X[1][l] : = convert(vectors(A)[1], matrix) : X[2][l] := convert(vectors(A)[2], matrix) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "X[1][lambda] \+ = matrix(X[1][l]) ; X[2][lambda] = matrix(X[2][l]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%\"XG6#\"\"\"6#%'lambdaG-%'matrixG6#7%7#!\"#7#F (7#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%\"XG6#\"\"#6#%'lambdaG -%'matrixG6#7%7#!\"$7#\"\"!7#\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "(e) Let a linear combin ation of the two eigenvectors be of the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "X[lambda] = mu[1] * X[1][lambda] + mu[2] * X[2][ lambda] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#%'lambdaG,&*&&%# muG6#\"\"\"F-&&F%F,F&F-F-*&&F+6#\"\"#F-&&F%F2F&F-F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Substit uting both eigenvectors with their respective numerical elements yield s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "X[lambda] = mu[1] * ma trix(X[1][l]) + mu[2] * matrix(X[2][l]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#%'lambdaG,&*&&%#muG6#\"\"\"F--%'matrixG6#7%7#! \"#7#F-7#\"\"!F-F-*&&F+6#\"\"#F--F/6#7%7#!\"$F5F4F-F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "E valuation of " }{XPPEDIT 18 0 "X[lambda]" "6#&%\"XG6#%'lambdaG" } {TEXT -1 7 " gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "X[l] := evalm(mu[1]*X[1][l] + mu[2]*X[2][l]) : X[lambda] = matrix(X[l]) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#%'lambdaG-%'matrixG6#7%7 #,&&%#muG6#\"\"\"!\"#*&\"\"$F1&F/6#\"\"#F1!\"\"7#F.7#F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "( f) Let the unit matrix [" }{TEXT 466 1 "U" }{TEXT -1 38 "] appropriate ly sized for this case be" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "U := diag(1,1,1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "( g) Substituting the respective matrices and eigenvalue into " }{TEXT 633 1 "(" }{TEXT -1 1 "[" }{TEXT 468 1 "A" }{TEXT -1 1 "]" }{TEXT 634 3 " \226 " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" } {TEXT 469 1 "U" }{TEXT -1 1 "]" }{TEXT 635 1 ")" }{TEXT -1 1 " " } {XPPEDIT 18 0 "X[lambda]" "6#&%\"XG6#%'lambdaG" }{TEXT -1 7 " gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "'(A - lambda*U)' * X[lamb da] = (matrix(A) - l * matrix(U)) * matrix(X[l]) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&,&%\"AG\"\"\"*&%'lambdaGF'%\"UGF'!\"\"F'&%\"XG6#F) F'*&,&-%'matrixG6#7%7%!\"#\"\"#\"\"$7%F7F'\"\"'7%F8F:F:F'*&F8F'-F26#7% 7%F'\"\"!FA7%FAF'FA7%FAFAF'F'F'F'-F26#7%7#,&&%#muG6#F'F6*&F8F'&FJ6#F7F 'F+7#FI7#FMF'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Evaluation of " }{TEXT 636 1 "(" }{TEXT -1 1 "[" }{TEXT 471 1 "A" }{TEXT -1 1 "]" }{TEXT 638 3 " \226 " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 472 1 "U " }{TEXT -1 1 "]" }{TEXT 637 1 ")" }{TEXT -1 1 " " }{XPPEDIT 18 0 "X[l ambda]" "6#&%\"XG6#%'lambdaG" }{TEXT -1 14 " yields the " }{TEXT 773 4 "zero" }{TEXT -1 24 " column matrix (vector)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "(matrix(A) - l * matrix(U)) * matrix(X[l] ) = evalm((A - l * U) &* X[l]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &,&-%'matrixG6#7%7%!\"#\"\"#\"\"$7%F,\"\"\"\"\"'7%F-F0F0F/*&F-F/-F'6#7 %7%F/\"\"!F77%F7F/F77%F7F7F/F/F/F/-F'6#7%7#,&&%#muG6#F/F+*&F-F/&F@6#F, F/!\"\"7#F?7#FCF/-F'6#7%7#F7FKFK" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "irrespective of numerical \+ values of the coefficients " }{XPPEDIT 18 0 "mu[1]" "6#&%#muG6#\"\"\" " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "mu[2]" "6#&%#muG6#\"\"#" } {TEXT 470 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 467 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 489 4 "N.B." }{TEXT -1 138 " \+ Raising a matrix to a positive or negative integer power does not chan ge its eigenvectors, but raises its eigenvalues to the same power." }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 641 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 491 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 642 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 490 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := matrix(2, 2 , [5, 8, 1, 7]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%\"AG-%'matrixG6#7$7$\"\"&\"\")7$\"\"\"\"\"(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Let the posi tive and negative exponents be" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "p := 3 : n := -2 : 'p' = p ; 'n' = n ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"pG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"n G!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenvalues of [" }{TEXT 492 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$\"\"*\"\"$" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "whence" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "root1(A) := charroots(A)[1] \+ : root2(A) := charroots(A)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1](A) = root1(A) ; char_root[2](A) = root 2(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\" AG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"#6#%\" AG\"\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(b) The" }{TEXT 493 1 " " }{TEXT -1 62 "sequence of l ists containing eigenvalues and eigenvectors of [" }{TEXT 494 1 "A" } {TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`roots& vectors(A)` := eigenvects(A) : roots_and_vectors(A) = `roots&vectors (A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6#%\"A G6$7%\"\"$\"\"\"<#-%'vectorG6#7$!\"%F+7%\"\"*F+<#-F.6#7$\"\"#F+" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "(c) The individual eigenvalues and corresponding eigenvectors a re" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "e[1] := `roots&vector s(A)`[1][1] : e[2] := `roots&vectors(A)`[2][1] :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 670 1 "\225" } {TEXT -1 76 " pair consisting of the first eigenvalue and the corresp onding eigenvector:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "if \+ e[1] = root1(A) then l := root1(A) : `vector(A)` := op(`roots&vec tors(A)`[1][3]) else l := e[2] : `vector(A)` := op(`roots&vectors( A)`[2][3]) fi :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "l1 := l : lambda[1] = l1 ; char_vector[lambda[1]](A) = op(`vector(A)`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%,char_vectorG6#&%'lambdaG6#\"\"\"6#%\"AG -%'vectorG6#7$\"\"#F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "or, as a column matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "X[l1] := convert(`vector( A)`, matrix) : X[lambda[1]] = matrix(X[l1]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"\"-%'matrixG6#7$7#\"\"#7#F*" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 671 1 "\225" }{TEXT -1 77 " pair consisting of the second eigen value and the corresponding eigenvector:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "if e[2] = root2(A) then l := root2(A) : `vector (A)` := op(`roots&vectors(A)`[2][3]) else l := e[1] : `vector(A)` \+ := op(`roots&vectors(A)`[1][3]) fi :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "l2:=l : lambda[2]=l2 ; char_vector[lambda[2]](A) = op(`vector(A)`) ; l:='l' : e[1]:='e[1]' : e[2]:='e[2]':" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%,char_vectorG6#&%'lambdaG6#\"\"#6#%\"AG-%'vectorG6# 7$!\"%\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "or, as a column matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "X[l2] := convert(`vector(A)`, matri x) : X[lambda[2]] = matrix(X[l2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"#-%'matrixG6#7$7#!\"%7#\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "( d) Raising the matrix [" }{TEXT 495 1 "A" }{TEXT -1 16 "] to the power " }{XPPEDIT 18 0 "p=3" "6#/%\"pG\"\"$" }{TEXT -1 30 " yields the fo llowing matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`A^3` := evalm(A^3) : A^3 = matrix(`A^3`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"$\"\"\"-%'matrixG6#7$7$\"$h#\"$O*7$\"$<\"\"$&\\" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(e) The eigenvalues of [" }{TEXT 496 1 "A" }{TEXT -1 1 "]" } {TEXT 497 1 "^" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 5 " are" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "charroots(A^3) := eigenvals( `A^3`) : char_roots(A^3) = charroots(A^3) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#*$)%\"AG\"\"$\"\"\"6$\"$H(\"#F" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "whence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "root1(A^3) \+ := charroots(A^3)[1] : root2(A^3) := charroots(A^3)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "char_root[1](A^3) = root1(A^3) ; \+ char_root[2](A^3) = root2(A^3) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- &%*char_rootG6#\"\"\"6#*$)%\"AG\"\"$F(\"$H(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"#6#*$)%\"AG\"\"$\"\"\"\"#F" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "For a comparison, the eigenvalues of [" }{TEXT 504 1 "A" } {TEXT -1 23 "] raised to the power " }{XPPEDIT 18 0 "p=3" "6#/%\"pG\" \"$" }{TEXT -1 5 " are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " [char_root[1](A)]^p = root1(A)^p ; [char_root[2](A)]^p = root2(A)^p \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)7#-&%*char_rootG6#\"\"\"6#%\" AG\"\"$F+\"$H(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)7#-&%*char_rootG 6#\"\"#6#%\"AG\"\"$\"\"\"\"#F" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "It can be seen from this comp arison that either eigenvalue of [" }{TEXT 498 1 "A" }{TEXT -1 1 "]" }{TEXT 499 1 "^" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 50 " is the c ube of the corresponding eigenvalue of [" }{TEXT 500 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(f) The" }{TEXT 501 1 " " }{TEXT -1 63 "sequence of lis ts containing eigenvalues and eigenvectors of [" }{TEXT 502 1 "A" } {TEXT -1 1 "]" }{TEXT 503 1 "^" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "`roots&vector s(A^3)` := eigenvects(`A^3`) : roots_and_vectors(A^3) = `roots&vecto rs(A^3)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6# *$)%\"AG\"\"$\"\"\"6$7%\"#FF+<#-%'vectorG6#7$!\"%F+7%\"$H(F+<#-F16#7$ \"\"#F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "It follows from the inspection of the above that the eigenvectors of [" }{TEXT 505 1 "A" }{TEXT -1 1 "]" }{TEXT 506 1 "^ " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 35 " are the same as those o f matrix [" }{TEXT 507 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(g) Raising the matrix [" }{TEXT 508 1 "A" }{TEXT -1 16 "] to the power " }{XPPEDIT 18 0 "n=-2" "6#/%\"nG,$\"\"#!\"\"" }{TEXT -1 30 " yields the followin g matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "`A^(-2)` := ev alm(A^(-2)) : A^` -2` = matrix(`A^(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-2G-%'matrixG6#7$7$#\"#>\"$V##!#KF.7$#!\"%F.# \"#6F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(h) The eigenvalues of [" }{TEXT 509 1 "A" }{TEXT -1 1 "]" }{TEXT 510 2 "^(" }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" } {TEXT 511 1 ")" }{TEXT -1 5 " are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "charroots(A^(-2)) := eigenvals(`A^(-2)`) : char_roo ts(A^` -2`) = charroots(A^(-2)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%+char_rootsG6#)%\"AG%$~-2G6$#\"\"\"\"\"*#F,\"#\")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "whence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "root1(A^(-2)) := charroot s(A^(-2))[1] : root2(A^(-2)) := charroots(A^(-2))[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "char_root[1](A^` -2`) = root1(A^(-2 )) ; char_root[2](A^` -2`) = root2(A^(-2)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#)%\"AG%$~-2G#F(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"#6#)%\"AG%$~-2G#\"\"\" \"#\")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "For a comparison, the eigenvalues of [" }{TEXT 512 1 "A" }{TEXT -1 23 "] raised to the power " }{XPPEDIT 18 0 "n=-2" "6# /%\"nG,$\"\"#!\"\"" }{TEXT -1 5 " are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "'[char_root[1](A)]'^` -2` = root1(A)^n ; '[char_roo t[2](A)]'^` -2` = root2(A)^n ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)7# -&%*char_rootG6#\"\"\"6#%\"AG%$~-2G#F*\"#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)7#-&%*char_rootG6#\"\"#6#%\"AG%$~-2G#\"\"\"\"\"*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "It can be seen from this comparison that either eigenvalue of \+ [" }{TEXT 514 1 "A" }{TEXT -1 1 "]" }{TEXT 515 2 "^(" }{XPPEDIT 18 0 " -2" "6#,$\"\"#!\"\"" }{TEXT 516 1 ")" }{TEXT -1 70 " is the square of the reciprocal of the corresponding eigenvalue of [" }{TEXT 513 1 "A " }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(i) The" }{TEXT 517 1 " " }{TEXT -1 63 "se quence of lists containing eigenvalues and eigenvectors of [" }{TEXT 518 1 "A" }{TEXT -1 1 "]" }{TEXT 519 2 "^(" }{XPPEDIT 18 0 "-2" "6#,$ \"\"#!\"\"" }{TEXT 520 1 ")" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 106 "`roots&vectors(A^(-2))` := eigenvects(`A^(-2) `) : roots_and_vectors(A^` -2`) = `roots&vectors(A^(-2))` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6#)%\"AG%$~-2G6$ 7%#\"\"\"\"#\")F-<#-%'vectorG6#7$\"\"#F-7%#F-\"\"*F-<#-F16#7$!\"%F-" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "It follows from the inspection of the above that the eige nvectors of [" }{TEXT 522 1 "A" }{TEXT -1 1 "]" }{TEXT 523 2 "^(" } {XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT 524 1 ")" }{TEXT -1 35 " a re the same as those of matrix [" }{TEXT 521 1 "A" }{TEXT -1 2 "]." }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 525 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 527 4 "N.B." }{TEXT -1 6 " If " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 32 " is an eigenvalue of a matrix \+ [" }{TEXT 526 1 "A" }{TEXT -1 34 "] with corresponding eigenvector " }{XPPEDIT 18 0 "X[lambda]" "6#&%\"XG6#%'lambdaG" }{TEXT 528 1 "," } {TEXT -1 23 " then for any scalar " }{XPPEDIT 18 0 "mu" "6#%#muG" } {TEXT 529 1 "," }{TEXT -1 2 " " }{XPPEDIT 18 0 "X[lambda]" "6#&%\"XG6 #%'lambdaG" }{TEXT -1 25 " is an eigenvector of [" }{TEXT 534 1 "A" }{TEXT -1 1 "]" }{TEXT 531 3 " \226 " }{XPPEDIT 18 0 "mu" "6#%#muG" } {TEXT 532 1 " " }{TEXT -1 1 "[" }{TEXT 533 1 "U" }{TEXT -1 36 "] corr esponding to the eigenvalue " }{XPPEDIT 18 0 "lambda - mu" "6#,&%'lam bdaG\"\"\"%#muG!\"\"" }{TEXT 530 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, cons ider a " }{TEXT 643 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 536 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 644 1 ")" }{TEXT -1 10 " \+ matrix [" }{TEXT 535 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A := matrix(2, 2, [-1, -5, 2, 6]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$ !\"\"!\"&7$\"\"#\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenvalues of [" }{TEXT 537 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$\"\"%\"\" \"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "whence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "root1 (A) := charroots(A)[1] : root2(A) := charroots(A)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1](A) = root1(A) ; char _root[2](A) = root2(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_ rootG6#\"\"\"6#%\"AG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char _rootG6#\"\"#6#%\"AG\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(b) The" }{TEXT 538 1 " " }{TEXT -1 62 "sequence of lists containing eigenvalues and eigenvectors of [ " }{TEXT 539 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`roots&vectors(A)` := eigenvects(A) : roots_and_vec tors(A) = `roots&vectors(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2 roots_and_vectorsG6#%\"AG6$7%\"\"%\"\"\"<#-%'vectorG6#7$F+!\"\"7%F+F+< #-F.6#7$#!\"&\"\"#F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "(c) Choosing and extracting, for i nstance, the smaller eigenvalue of [" }{TEXT 543 1 "A" }{TEXT -1 6 "] \+ give" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "e[1](A) := `roots&v ectors(A)`[1][1] : e[2](A) := `roots&vectors(A)`[2][1] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "l(A) := min(e[1](A), e[2](A)) : l ambda[A] = l(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#%\"A G\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "(d) The corresponding eigenvector of [" }{TEXT 551 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "if e[1](A) = l(A) then `vector(A)` := op(`roots&vectors(A)`[1][3]) else `vector(A)` := op(`roots&vectors(A)`[2][3]) fi : char_vecto r[lambda](A) = op(`vector(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- &%,char_vectorG6#%'lambdaG6#%\"AG-%'vectorG6#7$#!\"&\"\"#\"\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "or, as a column matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "X[l(A)] := convert(`vector(A)`, matrix) : X[lambda[ A]] = matrix(X[l(A)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&% 'lambdaG6#%\"AG-%'matrixG6#7$7##!\"&\"\"#7#\"\"\"" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(e) Let th e unit matrix [" }{TEXT 540 1 "U" }{TEXT -1 38 "] appropriately sized \+ for this case be" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "U := di ag(1,1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG- %'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "(f) Let a " }{TEXT 645 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 541 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 646 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 542 1 "B" }{TEXT -1 15 "] be defined as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B := A - mu*U : 'B' = B ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG,&%\"AG\"\"\"*&%#muGF'%\"UGF'!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "S ubstituting [" }{TEXT 544 1 "A" }{TEXT -1 7 "] and [" }{TEXT 545 1 "U " }{TEXT -1 46 "] with their numerical values and evaluating [" } {TEXT 546 1 "B" }{TEXT -1 7 "] yield" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B := evalm(B) : B = matrix(B) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$,&!\"\"\"\"\"%#muGF+!\"&7$\"\"# ,&\"\"'F,F-F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(g) The eigenvalues of [" }{TEXT 547 1 "B " }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "cha rroots(B) := eigenvals(B) : char_roots(B) = charroots(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"BG6$,&%#muG!\"\"\"\"\"F, ,&F*F+\"\"%F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "whence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "root1(B) := charroots(B)[1] : root2(B) := charroots (B)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1](B) = root1(B) ; char_root[2](B) = root2(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\"BG,&%#muG!\"\"F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"#6#%\"BG,&%#muG! \"\"\"\"%\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(h) The" }{TEXT 548 1 " " }{TEXT -1 62 "se quence of lists containing eigenvalues and eigenvectors of [" }{TEXT 549 1 "B" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`roots&vectors(B)` := eigenvects(B) : roots_and_vectors(B) = `ro ots&vectors(B)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vec torsG6#%\"BG6$7%,&%#muG!\"\"\"\"\"F-F-<#-%'vectorG6#7$#!\"&\"\"#F-7%,& F+F,\"\"%F-F-<#-F06#7$F,F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "(i) Extracting the smaller eigenv alue of [" }{TEXT 550 1 "B" }{TEXT -1 7 "] gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "e[1](B) := `roots&vectors(B)`[1][1] : e[2]( B) := `roots&vectors(B)`[2][1] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "l(B) := min(e[1](B), e[2](B)) : lambda[B] = l(B) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#%\"BG,&%#muG!\"\"\"\" \"F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 20 "which is the same as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "lambda[A] - mu = l(A) - mu ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%'lambdaG6#%\"AG\"\"\"%#muG!\"\",&F*F+F)F)" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "(j) The corresponding eigenvector of [" }{TEXT 552 1 "B" } {TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "if e[ 1](B) = l(B) then `vector(B)` := op(`roots&vectors(B)`[1][3]) else \+ `vector(B)` := op(`roots&vectors(B)`[2][3]) fi : char_vector[lambd a](B) = op(`vector(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%,char _vectorG6#%'lambdaG6#%\"BG-%'vectorG6#7$#!\"&\"\"#\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "o r, as a column matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "X[l(B)] := convert(`vector(B)`, matrix) : X[lambda[B]] = mat rix(X[l(B)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6 #%\"BG-%'matrixG6#7$7##!\"&\"\"#7#\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "which is the same as " }{XPPEDIT 18 0 "X[lambda[A]]" "6#&%\"XG6#&%'lambdaG6#%\"AG" } {TEXT 553 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 554 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 417 4 "N.B." }{TEXT -1 61 " A square matrix and its transpose have the same eigenvalues" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 420 10 "char_roots" }{TEXT -1 1 "[" }{TEXT 418 1 "A" }{TEXT -1 1 "]" }{TEXT 647 20 " = char_roots(Transp" }{TEXT -1 1 "[" }{TEXT 419 1 "A" }{TEXT -1 1 "]" }{TEXT 421 1 ")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "F or example, consider a " }{TEXT 648 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"# " }{TEXT 423 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 649 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 422 1 "A" }{TEXT -1 10 "] given as" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A := matrix(2, 2, [3, 5, - 2, -4]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG- %'matrixG6#7$7$\"\"$\"\"&7$!\"#!\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenvalues of \+ [" }{TEXT 424 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = char roots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$ \"\"\"!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "(b) The eigenvalues of the transpose of [" } {TEXT 425 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "charroots(transp(A)) := eigenvals(transpose(A)) : c har_roots(Transp(A)) = charroots(transp(A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#-%'TranspG6#%\"AG6$\"\"\"!\"#" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 426 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 716 4 "N.B." }{TEXT -1 6 " If " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 44 " is an eigenvalue of an invert ible matrix [" }{TEXT 723 1 "A" }{TEXT -1 9 "], then " }{XPPEDIT 18 0 "lambda^`-1`" "6#)%'lambdaG%#-1G" }{TEXT -1 38 " is an eigenvalue o f the inverse of [" }{TEXT 715 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For exa mple, consider a " }{TEXT 718 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 721 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 722 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 717 1 "A" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A := matrix(3, 3, [2, 10, - 2, 10, 5, 8, -2, 8, 11]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"#5!\"#7%F+\"\"&\"\")7%F,F/ \"#6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 24 "(a) The eigenvalues of [" }{TEXT 719 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := e igenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6%\"\"*!\"*\"#=" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "(b) The eigenvalues of the inverse of [" }{TEXT 720 1 "A" }{TEXT -1 5 "] are " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "charroots(inv(A)) := ei genvals(inverse(A)) : char_roots(Inv(A)) = charroots(inv(A)) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#-%$InvG6#%\"AG6%#\"\" \"\"\"*#!\"\"F.#F-\"#=" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 258 "" 0 "" {TEXT 724 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 589 4 "N.B." }{TEXT -1 40 " The eigenvalues and determinant of a " }{TEXT 588 9 "Hermiti an" }{TEXT -1 18 " matrix are real." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 650 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 594 3 " \327 \+ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 651 1 ")" }{TEXT -1 2 " " } {TEXT 591 9 "Hermitian" }{TEXT -1 10 " matrix [" }{TEXT 590 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " A := matrix(2, 2, [2, 3+4*I, 3-4*I, -5]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"#^$\"\"$\"\"%7$^$F ,!\"%!\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 20 "The eigenvalues of [" }{TEXT 593 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$,&#!\"$\"\"#\"\"\"*&#F-F,F--%%s qrtG6#\"$\\\"F-F-,&F*F-*&#F-F,F-*$F0F-F-!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The determinan t of [" }{TEXT 738 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG!#N" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "[ For a d efinition of the " }{TEXT 706 9 "Hermitian" }{TEXT -1 31 " matrix, r efer to Unit (16). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 258 "" 0 "" {TEXT 592 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 427 4 "N.B." }{TEXT -1 24 " The eigenvalues of a " }{TEXT 429 8 "diagonal" }{TEXT -1 74 " matrix of any order are equal to the elements on the principal diag onal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 652 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 430 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 653 1 ")" }{TEXT -1 19 " diagonal matrix [" }{TEXT 428 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " A := diag(3, 0, 2, 5) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&\"\"$\"\"!F+F+7&F+F+F+F+7&F+F+\" \"#F+7&F+F+F+\"\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The eigenvalues of [" }{TEXT 431 1 "A" } {TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charro ots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6&\"\"$\"\"!\"\"#\"\"&" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 432 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 433 4 "N.B." }{TEXT -1 62 " The sum of the eige nvalues of a matrix is equal to its trace" }}}{EXCHG {PARA 258 "" 0 " " {XPPEDIT 18 0 "Sum(char_root[i](A),i=1..n)" "6#-%$SumG6$-&%*char_roo tG6#%\"iG6#%\"AG/F*;\"\"\"%\"nG" }{TEXT -1 1 " " }{TEXT 440 1 "=" } {TEXT -1 1 " " }{TEXT 441 5 "Trace" }{TEXT -1 1 "[" }{TEXT 442 1 "A" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 654 1 " (" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 435 3 " \327 " }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 655 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 434 1 " A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A := matrix(3, 3, [5, 2, 2, 3, 6, 3, 6, 6, 9]) : A = matrix(A) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"&\"\"#F+ 7%\"\"$\"\"'F-7%F.F.\"\"*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenvalues of [" }{TEXT 437 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6%\"#9\"\"$F *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "(b) The sum of the eigenvalues of [" }{TEXT 436 1 "A" } {TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "No_root s(A) := nops([charroots(A)]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`sum(charroots(A))` := sum(charroots(A)[i], i=1..No_roots(A)) \+ :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Sum(char_root[i](A), i =1..No_roots(A)) = `sum(charroots(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$-&%*char_rootG6#%\"iG6#%\"AG/F+;\"\"\"\"\"$\"#?" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(c) The trace of [" }{TEXT 438 1 "A" }{TEXT -1 4 "] is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`trace(A)` := trace(A) : T race(A) = `trace(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6# %\"AG\"#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 439 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 443 4 "N.B." }{TEXT -1 93 " The pro duct of the eigenvalues (counting multiplicities) of a matrix equals i ts determinant" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Product(cha r_root[i](A), i=1..n)" "6#-%(ProductG6$-&%*char_rootG6#%\"iG6#%\"AG/F* ;\"\"\"%\"nG" }{TEXT 656 6 " = Det" }{TEXT -1 1 "[" }{TEXT 444 1 "A" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "For example, consider the same " } {TEXT 657 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 446 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 658 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 445 1 "A" }{TEXT -1 12 "] as before." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "(a) The produc t of the eigenvalues of [" }{TEXT 447 1 "A" }{TEXT -1 4 "] is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`product(charroots(A))` := p roduct(charroots(A)[i], i=1..No_roots(A)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Product(char_root[i](A), i=1..No_roots(A)) = `prod uct(charroots(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$ -&%*char_rootG6#%\"iG6#%\"AG/F+;\"\"\"\"\"$\"$E\"" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(b) The de terminant of [" }{TEXT 448 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"$E\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 669 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 " " 0 "" {TEXT -1 3 "B. " }{TEXT 555 27 "The Cayley-Hamilton theorem" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 586 15 "Cayley-Hamilton" }{TEXT -1 30 " theorem ( also known as the " }{TEXT 587 15 "Hamilton-Cayley" }{TEXT -1 163 " \+ theorem) states that every square matrix satisfies its own characteris tic equation. This means that, if the characteristic determinant assoc iated with a matrix [" }{TEXT 563 1 "A" }{TEXT -1 30 "] is expanded to take the form" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 558 4 "Det(" }{TEXT -1 1 "[" }{TEXT 556 1 "A" }{TEXT -1 1 "]" }{TEXT 659 3 " \226 " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 2 " [" }{TEXT 557 1 "U " }{TEXT -1 1 "]" }{TEXT 559 4 ") = " }{XPPEDIT 18 0 "b[n]*lambda^n+b[ n-1]*lambda^(n-1)" "6#,&*&&%\"bG6#%\"nG\"\"\")%'lambdaGF(F)F)*&&F&6#,& F(F)F)!\"\"F))F+,&F(F)F)F0F)F)" }{TEXT 560 9 " + ... + " }{XPPEDIT 18 0 "b[2]*lambda^2+b[1]*lambda +b[0]" "6#,(*&&%\"bG6#\"\"#\"\"\"*$%'lamb daGF(F)F)*&&F&6#F)F)F+F)F)&F&6#\"\"!F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "then" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "b[n]*A^n+b[n-1]*A^(n-1)" "6#,&*&&%\"b G6#%\"nG\"\"\")%\"AGF(F)F)*&&F&6#,&F(F)F)!\"\"F))F+,&F(F)F)F0F)F)" } {TEXT 561 9 " + ... + " }{XPPEDIT 18 0 "b[2]*A^2+b[1]*A + b[0]" "6#,(* &&%\"bG6#\"\"#\"\"\"*$%\"AGF(F)F)*&&F&6#F)F)F+F)F)&F&6#\"\"!F)" } {TEXT 567 1 " " }{TEXT -1 1 "[" }{TEXT 568 1 "U" }{TEXT -1 1 "]" } {TEXT 562 3 " = " }{TEXT -1 1 "[" }{TEXT 564 1 "0" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 660 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 566 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 661 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 565 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A := matrix(3, 3 , [5, 2, 2, 3, 6, 3, 6, 6, 9]) : A = matrix(A) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"&\"\"#F+7%\"\"$\"\"'F-7%F.F .\"\"*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "(a) The characteristic equation of [" }{TEXT 569 1 " A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "cha r_eq(A) := charpoly(A, lambda) = 0 : char_eq(A) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,**$)%'lambdaG\"\"$\"\"\"F)*&\"#?F))F'\"\"#F)!\"\"*& \"#$*F)F'F)F)\"$E\"F.\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) Substituting [" }{TEXT 570 1 "A" }{TEXT -1 7 "] for " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" } {TEXT -1 8 " yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Eq_ A := subs(lambda=A, char_eq(A)) : Eq_A ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"AG\"\"$\"\"\"F)*&\"#?F))F'\"\"#F)!\"\"*&\"#$*F )F'F)F)\"$E\"F.\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 260 "" 1 "" {TEXT -1 87 "(c) Multiplying the last term of the left-hand side of this equation by a unit matrix [" }{TEXT 571 1 "U" }{TEXT -1 34 "] gives a matrix equation denoted " }{TEXT 574 4 "Eq _A" }{TEXT -1 55 ", which corresponds to the characteristic equation o f [" }{TEXT 573 1 "A" }{TEXT -1 7 "], viz." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 79 "s := 0 : for i to nops(lhs(Eq_A)) - 1 do s : = s + op(i, lhs(Eq_A)) od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Eq_A := s + op(nops(lhs(Eq_A)), lhs(Eq_A)) * U = 0 : Eq_A ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"AG\"\"$\"\"\"F)*&\"#?F))F'\" \"#F)!\"\"*&\"#$*F)F'F)F)*&\"$E\"F)%\"UGF)F.\"\"!" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(d) The un it matrix [" }{TEXT 572 1 "U" }{TEXT -1 38 "] appropriately sized for \+ this case is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U := diag(1 , 1, 1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG- %'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(e) Substitutin g [" }{TEXT 575 1 "A" }{TEXT -1 7 "] and [" }{TEXT 576 1 "U" }{TEXT -1 35 "] with their numerical values into " }{TEXT 577 4 "Eq_A" } {TEXT -1 39 " and evaluating its left-hand side give" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "lhs(Eq_A) = evalm(lhs(Eq_A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"AG\"\"$\"\"\"F)*&\"#?F))F'\"\"#F)! \"\"*&\"#$*F)F'F)F)*&\"$E\"F)%\"UGF)F.-%'matrixG6#7%7%\"\"!F9F9F8F8" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "which verifies the " }{TEXT 585 15 "Cayley-Hamilton" } {TEXT -1 10 " theorem." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "[ Refer to Units (23) through (25 ) for the application of the theorem to computing functions of matrice s. ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 737 5 "* * *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (22) for \"" }{TEXT 668 22 "Similarity of matrices" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 667 67 "-------------------------------------- -----------------------------" }}}}{MARK "7 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }