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MATRIX OPE RATIONS: Unit 22" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 259 "" 0 "" {TEXT 262 23 "Dr. Wlodzislaw Kostecki" }}{PARA 260 "" 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 262 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 263 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 261 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 265 "" 0 "" {TEXT 263 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT 264 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(22)" }{TEXT 259 1 " " }{TEXT 258 22 "Similarity of mat rices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 609 10 "OBJECTIVES" }{TEXT 610 1 ":" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 611 1 "\225" }{TEXT -1 62 " To state the condition necessary for square matrices to be \+ " }{TEXT 612 7 "similar" }{TEXT -1 57 " and express it in the form of two equivalent relations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 613 1 " \225" }{TEXT -1 42 " To provide examples of similar matrices." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 614 1 "\225" }{TEXT -1 28 " To introduce the function " }{TEXT 615 9 "issimilar" }{TEXT -1 37 " for testing ma trices for similarity." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 616 1 "\225" } {TEXT -1 77 " To investigate properties of certain operations involvi ng similar matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 617 1 "\225" } {TEXT -1 17 " To define the " }{TEXT 618 5 "modal" }{TEXT -1 9 " ma trix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 620 1 "\225" }{TEXT -1 75 " To state the condition necessary for the eigenvectors of a matrix to be \+ " }{TEXT 619 20 "linearly independent" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 649 1 "\225" }{TEXT -1 86 " To propose a univer sal method of extracting eigenvectors and associating them with " } {TEXT 650 8 "distinct" }{TEXT -1 32 " eigenvalues of a matrix in a \+ " }{TEXT 651 6 "unique" }{TEXT -1 6 " way." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 621 1 "\225" }{TEXT -1 91 " To introduce the term 'unique' (e nclosed in single quotes) for use in connection with a " }{TEXT 623 5 "modal" }{TEXT -1 41 " matrix whose columns are eigenvectors " } {TEXT 624 8 "uniquely" }{TEXT -1 19 " associated with " }{TEXT 625 8 "distinct" }{TEXT -1 32 " eigenvalues of a given matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 626 1 "\225" }{TEXT -1 189 " To suggest \+ an application of the 'unique' modal matrix to computing functions of \+ matrices and stress that permutations of elements in modal-matrix colu mns do not affect such computations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 628 1 "\225" }{TEXT -1 96 " To present several variants of a 'unique' modal matrix associated with a given matrix having " }{TEXT 629 8 "d istinct" }{TEXT -1 14 " eigenvalues." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 631 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 630 6 "Jord an" }{TEXT -1 26 " form of a square matrix." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 632 1 "\225" }{TEXT -1 25 " To emphasise that the " }{TEXT 633 6 "Jordan" }{TEXT -1 35 " form of a square matrix must be " } {TEXT 636 6 "unique" }{TEXT -1 38 " for computation of matrix functio ns." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 634 1 "\225" }{TEXT -1 33 " To p oint out that the function " }{TEXT 635 6 "jordan" }{TEXT -1 20 " does not return a " }{TEXT 637 6 "unique" }{TEXT -1 62 " matrix and prov ide alternative methods of constructing the " }{TEXT 639 6 "unique" } {TEXT -1 2 " " }{TEXT 638 6 "Jordan" }{TEXT -1 26 " form of a square matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 642 1 "\225" }{TEXT -1 80 " \+ To provide equivalent expressions of a statement that every square ma trix is " }{TEXT 640 7 "similar" }{TEXT -1 21 " to the associated \+ " }{TEXT 641 6 "Jordan" }{TEXT -1 66 " form of the matrix and vice ve rsa, and give supporting examples." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 646 1 "\225" }{TEXT -1 82 " To discuss application of matrix similari ty for computing functions of matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 647 1 "\225" }{TEXT -1 110 " To provide rules for computation o f functions of diagonal and non-diagonal matrices using matrix similar ity." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 648 1 "\225" }{TEXT -1 82 " To \+ provide alternative methods of computation of functions of diagonal ma trices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 152 "restart : interface(warnlevel=0) : with(l inalg, augment, charpoly, coldim, diag, eigenvals, eigenvects, inverse , issimilar, jordan, rowdim, trace) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 428 3 "A. " }{TEXT 429 16 "Similar matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Two square matrices of the same or der, [" }{TEXT 424 1 "A" }{TEXT -1 7 "] and [" }{TEXT 425 1 "B" } {TEXT -1 8 "], are " }{TEXT 270 7 "similar" }{TEXT -1 52 " if and on ly if there exists an invertible matrix [" }{TEXT 269 1 "S" }{TEXT -1 11 "] such that" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 1 "[" }{TEXT 266 1 "A" }{TEXT -1 1 "]" }{TEXT 265 6 " = Inv" }{TEXT -1 1 "[" } {TEXT 271 1 "S" }{TEXT -1 3 "] [" }{TEXT 272 1 "B" }{TEXT -1 3 "] [" } {TEXT 273 1 "S" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "If [" }{TEXT 483 1 "A" }{TEXT -1 17 "] is similar to [" }{TEXT 484 1 "B" }{TEXT -1 9 "], then [" } {TEXT 485 1 "B" }{TEXT -1 22 "] is also similar to [" }{TEXT 486 1 "A " }{TEXT -1 7 "], i.e." }}}{EXCHG {PARA 281 "" 0 "" {TEXT -1 1 "[" } {TEXT 488 1 "B" }{TEXT -1 1 "]" }{TEXT 487 3 " = " }{TEXT -1 1 "[" } {TEXT 489 1 "S" }{TEXT -1 3 "] [" }{TEXT 490 1 "A" }{TEXT -1 1 "]" } {TEXT 491 4 " Inv" }{TEXT -1 1 "[" }{TEXT 492 1 "S" }{TEXT -1 1 "]" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "A relationship equivalent to either of the above is" }}}{EXCHG {PARA 285 "" 0 "" {TEXT -1 1 "[" }{TEXT 541 1 "S" }{TEXT -1 3 "] [" } {TEXT 538 1 "A" }{TEXT -1 1 "]" }{TEXT 537 3 " = " }{TEXT -1 1 "[" } {TEXT 539 1 "B" }{TEXT -1 3 "] [" }{TEXT 540 1 "S" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Exemplarily, consider " }{TEXT 268 1 "(" }{XPPEDIT 18 0 "3" "6 #\"\"$" }{TEXT 274 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 275 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 267 1 "A" }{TEXT -1 7 "] and \+ [" }{TEXT 276 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "A := matrix(3, 3, [6, 1, 0, 0, 6, 1, 0, 0, 6]) \+ : B := matrix(3, 3, [6, 3, 0, 0, 6, 3, 0, 0, 6]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"'\"\"\"\"\"! 7%F,F*F+7%F,F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6# 7%7%\"\"'\"\"$\"\"!7%F,F*F+7%F,F,F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "and find out if matrice s [" }{TEXT 281 1 "A" }{TEXT -1 7 "] and [" }{TEXT 282 1 "B" }{TEXT -1 23 "] are mutually similar." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Let an invertible matrix [" } {TEXT 286 1 "S" }{TEXT -1 22 "] be given in the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "S := matrix(3, 3, [9, 0, 0, 0, 3, 0, 0, 0 , 1]) : S = matrix(S) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"SG-%' matrixG6#7%7%\"\"*\"\"!F+7%F+\"\"$F+7%F+F+\"\"\"" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(1) The pr oduct " }{TEXT 277 3 "Inv" }{TEXT -1 1 "[" }{TEXT 278 1 "S" }{TEXT -1 3 "] [" }{TEXT 279 1 "B" }{TEXT -1 3 "] [" }{TEXT 280 1 "S" }{TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`inv(S) B S` := evalm(inverse(S) &* B &* S) : Inv(S)*B*S = matri x(`inv(S) B S`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%$InvG6#%\"SG \"\"\"%\"BGF)F(F)-%'matrixG6#7%7%\"\"'F)\"\"!7%F1F0F)7%F1F1F0" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "which is equal to matrix [" }{TEXT 283 1 "A" }{TEXT -1 22 "]. T herefore, matrix [" }{TEXT 284 1 "A" }{TEXT -1 36 "] is found to be si milar to matrix [" }{TEXT 285 1 "B" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This op eration may be presented in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "matrix(A) = inverse(S) * matrix(B) \+ * matrix(S) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7%\"\" '\"\"\"\"\"!7%F+F)F*7%F+F+F)*(-F%6#7%7%#F*\"\"*F+F+7%F+#F*\"\"$F+7%F+F +F*F*-F%6#7%7%F)F7F+7%F+F)F7F-F*-F%6#7%7%F4F+F+7%F+F7F+F8F*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(2) The product [" }{TEXT 493 1 "S" }{TEXT -1 3 "] [" }{TEXT 494 1 "A" }{TEXT -1 1 "]" }{TEXT 495 4 " Inv" }{TEXT -1 1 "[" }{TEXT 496 1 "S" }{TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`S A inv(S)` := evalm(S &* A &* inverse(S)) : \+ S*A*Inv(S) = matrix(`S A inv(S)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*(%\"SG\"\"\"%\"AGF&-%$InvG6#F%F&-%'matrixG6#7%7%\"\"'\"\"$\"\"!7%F2 F0F17%F2F2F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "which is equal to matrix [" }{TEXT 497 1 "B" } {TEXT -1 22 "]. Therefore, matrix [" }{TEXT 498 1 "B" }{TEXT -1 24 "] \+ is similar to matrix [" }{TEXT 499 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "T his operation may be presented in \"like-in-a-book\" form, viz." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "matrix(B) = matrix(S) * matr ix(A) * inverse(S) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7 %7%\"\"'\"\"$\"\"!7%F+F)F*7%F+F+F)*(-F%6#7%7%\"\"*F+F+7%F+F*F+7%F+F+\" \"\"F6-F%6#7%7%F)F6F+7%F+F)F6F-F6-F%6#7%7%#F6F3F+F+7%F+#F6F*F+F5F6" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "(3a) The matrix product [" }{TEXT 542 1 "S" }{TEXT -1 3 "] [" }{TEXT 543 1 "A" }{TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "`SA` := evalm(S &* A) : S*A = matrix(`S A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"SG\"\"\"%\"AGF&-%'matri xG6#7%7%\"#a\"\"*\"\"!7%F/\"#=\"\"$7%F/F/\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "(3b) The mat rix product [" }{TEXT 544 1 "B" }{TEXT -1 3 "] [" }{TEXT 545 1 "S" } {TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`BS` := evalm(B &* S) : `B S` = matrix(`BS`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%$B~SG-%'matrixG6#7%7%\"#a\"\"*\"\"!7 %F,\"#=\"\"$7%F,F,\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Both product matrices of (3a) and \+ (3b) are equal, which verifies that matrices [" }{TEXT 560 1 "A" } {TEXT -1 7 "] and [" }{TEXT 561 1 "B" }{TEXT -1 64 "] are similar. Thi s verification may be done directly using the " }{TEXT 562 9 "issimila r" }{TEXT -1 15 " function, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "issimilar(A, B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 652 7 "Boolean" }{TEXT -1 9 " value \+ " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 18 " returned by the " }{TEXT 563 9 "issimilar" }{TEXT -1 28 " function verifies the fact." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 287 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 4 "N.B." }{TEXT -1 113 " Similar matrices h ave the same characteristic equation and, therefore, the same eigenval ues and the same trace." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Exemplarily, consider the similar \+ matrices [" }{TEXT 289 1 "A" }{TEXT -1 7 "] and [" }{TEXT 290 1 "B" } {TEXT -1 16 "] defined above." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "(a) The characteristic equati on, eigenvalues, and trace of matrix [" }{TEXT 291 1 "A" }{TEXT -1 5 " ] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "l := lambda :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "char_eq(A) := charpoly(A, l) = 0 : charroots(A) := eigenvals(A) : `trace(A)` := trace(A) :" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "char_eq(A) ; char_roots(A ) = charroots(A) ; Trace(A) = `trace(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%'lambdaG\"\"\"\"\"'!\"\"\"\"$F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6%\"\"'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6#%\"AG\"#=" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "(b) The char acteristic equation, eigenvalues, and trace of matrix [" }{TEXT 292 1 "B" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "c har_eq(B) := charpoly(B, l) = 0 : charroots(B) := eigenvals(B) : ` trace(B)` := trace(B) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " char_eq(B) ; char_roots(B) = charroots(B) ; Trace(B) = `trace(B)` \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%'lambdaG\"\"\"\"\"'!\"\"\" \"$F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"BG6% \"\"'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6#%\"BG\"#=" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "It is easily observed that the results of (a) and (b) are ident ical." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 293 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 302 4 "N.B." }{TEXT -1 6 " If [" }{TEXT 300 1 "X" }{TEXT -1 33 "] is an eigenvector of a matrix [" }{TEXT 301 1 "A" }{TEXT -1 30 "] associated with eigenvalue " }{XPPEDIT 18 0 "la mbda" "6#%'lambdaG" }{TEXT -1 7 " and [" }{TEXT 299 1 "A" }{TEXT -1 26 "] is similar to a matrix [" }{TEXT 303 1 "B" }{TEXT -1 10 "], then [" }{TEXT 295 1 "Y" }{TEXT -1 1 "]" }{TEXT 294 3 " = " }{TEXT -1 1 " [" }{TEXT 296 1 "S" }{TEXT -1 3 "] [" }{TEXT 297 1 "X" }{TEXT -1 26 "] is an eigenvector for [" }{TEXT 298 1 "B" }{TEXT -1 39 "] correspond ing to the same eigenvalue." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Exemplarily, consider the simila r matrices [" }{TEXT 304 1 "A" }{TEXT -1 7 "] and [" }{TEXT 305 1 "B" }{TEXT -1 50 "] defined earlier and the same invertible matrix [" } {TEXT 311 1 "S" }{TEXT -1 28 "] with which the equality [" }{TEXT 307 1 "A" }{TEXT -1 1 "]" }{TEXT 306 6 " = Inv" }{TEXT -1 1 "[" } {TEXT 308 1 "S" }{TEXT -1 3 "] [" }{TEXT 309 1 "B" }{TEXT -1 3 "] [" } {TEXT 310 1 "S" }{TEXT -1 9 "] holds." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The eigenvalu es of [" }{TEXT 312 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)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "from which it can be seen that matrix [" }{TEXT 313 1 "A" }{TEXT -1 37 "] has one root of multiplicity three." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Denoting this root as " }{XPPEDIT 18 0 "lambda[A]" "6#&% 'lambdaG6#%\"AG" }{TEXT -1 56 " and extracting it from from the solut ion sequence give" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "root(A ) := charroots(A)[1] : l[A] = root(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#%\"AG\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(b) The" }{TEXT 314 1 " " }{TEXT -1 62 "sequence of lists containing eigenvalues and e igenvectors of [" }{TEXT 315 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`roots&vectors(A)` := eigenvects(A) : r oots_and_vectors(A) = `roots&vectors(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6#%\"AG7%\"\"'\"\"$<#-%'vectorG6# 7%\"\"\"\"\"!F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "(c) Extracting the set of the eigenvector s of [" }{TEXT 316 1 "A" }{TEXT -1 8 "] yields" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "vectors(A) := `roots&vectors(A)`[3] : char_v ectors(A) = vectors(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-char_v ectorsG6#%\"AG<#-%'vectorG6#7%\"\"\"\"\"!F." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "(d) Extracting \+ the eigenvector from the above set gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "`vector(A)` := vectors(A)[1] : char_vector[l](A) = \+ op(`vector(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%,char_vectorG 6#%'lambdaG6#%\"AG-%'vectorG6#7%\"\"\"\"\"!F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "or, as a colum n matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "X[l(A )] := convert(`vector(A)`, matrix) : X[l[A]] = matrix(X[l(A)]) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#%\"AG-%'matrixG6# 7%7#\"\"\"7#\"\"!F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(e) The eigenvalues of [" }{TEXT 317 1 "B" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(B) := eigenvals(B) : char_roots(B) = charroots(B) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"BG6%\"\"'F)F) " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "from which it can be seen that matrix [" }{TEXT 318 1 "B " }{TEXT -1 42 "] also has one root of multiplicity three." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "D enoting this root as " }{XPPEDIT 18 0 "lambda[B]" "6#&%'lambdaG6#%\"B G" }{TEXT -1 57 " and extracting it from from the solution sequence g ives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "root(B) := charroot s(B)[1] : l[B] = root(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'la mbdaG6#%\"BG\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(f) The" }{TEXT 319 1 " " }{TEXT -1 62 "se quence of lists containing eigenvalues and eigenvectors of [" }{TEXT 320 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#%\"BG7%\"\"'\"\"$<#-%'vectorG6#7%\"\"\"\"\"!F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "( g) Extracting the set of the eigenvectors of [" }{TEXT 321 1 "B" } {TEXT -1 8 "] yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "vec tors(B) := `roots&vectors(B)`[3] : char_vectors(B) = vectors(B) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-char_vectorsG6#%\"BG<#-%'vectorG6 #7%\"\"\"\"\"!F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "(h) Extracting the eigenvector from the a bove set gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "`vector(B )` := vectors(B)[1] : char_vector[l](B) = op(`vector(B)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%,char_vectorG6#%'lambdaG6#%\"BG-%' vectorG6#7%\"\"\"\"\"!F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "or, as a column matrix (vector)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "X[l(B)] := convert(`vecto r(B)`, matrix) : X[l[B]] = matrix(X[l(B)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#%\"BG-%'matrixG6#7%7#\"\"\"7#\"\"! F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(i) The product matrix [" }{TEXT 323 1 "Y" }{TEXT -1 1 " ]" }{TEXT 322 3 " = " }{TEXT -1 1 "[" }{TEXT 324 1 "S" }{TEXT -1 3 "] \+ [" }{TEXT 325 1 "X" }{TEXT -1 43 "] is the following column matrix (v ector):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Y := evalm(S &* \+ X[l(A)]) : Y = matrix(Y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"YG -%'matrixG6#7%7#\"\"*7#\"\"!F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(j) From the inspection of \+ " }{XPPEDIT 18 0 "X[lambda[B]]" "6#&%\"XG6#&%'lambdaG6#%\"BG" }{TEXT -1 8 " and [" }{TEXT 326 1 "Y" }{TEXT -1 33 "], it can be easily no ticed that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Y = 9*X[l[B]] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"YG,$&%\"XG6#&%'lambdaG6#%\"B G\"\"*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "matr ix(Y) = 9*matrix(X[l(B)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'mat rixG6#7%7#\"\"*7#\"\"!F*,$-F%6#7%7#\"\"\"F*F*F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "(k) Accordin g to the statement found in Unit (21), the product of the eigenvector \+ of [" }{TEXT 327 1 "B" }{TEXT -1 9 "], i.e. " }{XPPEDIT 18 0 "X[lambd a[B]]" "6#&%\"XG6#&%'lambdaG6#%\"BG" }{TEXT 328 2 " ," }{TEXT -1 33 " \+ and an arbitrary scalar, i.e. " }{XPPEDIT 18 0 "9" "6#\"\"*" }{TEXT -1 55 " in this particular case, is also the eigenvector of [" } {TEXT 330 1 "B" }{TEXT -1 15 "]. Therefore, [" }{TEXT 331 1 "Y" } {TEXT -1 25 "] is the eigenvector of [" }{TEXT 329 1 "B" }{TEXT -1 22 "]. This implies that [" }{TEXT 332 1 "Y" }{TEXT -1 136 "] is a soluti on to the equation (4) of Unit (21). Using the designations applicable to this particular case, equation (4) takes the form" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 336 1 "(" }{TEXT -1 1 "[" }{TEXT 338 1 "B" } {TEXT -1 1 "]" }{TEXT 333 3 " \226 " }{XPPEDIT 18 0 "lambda[B]" "6#&%' lambdaG6#%\"BG" }{TEXT -1 2 " [" }{TEXT 334 1 "U" }{TEXT -1 1 "]" } {TEXT 335 2 ") " }{TEXT -1 1 "[" }{TEXT 339 1 "Y" }{TEXT -1 1 "]" } {TEXT 337 3 " = " }{TEXT -1 1 "[" }{TEXT 340 1 "0" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "(l) With the unit matrix [" }{TEXT 341 1 "U" }{TEXT -1 41 "] ap propriately sized for this case, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U := diag(1, 1, 1) : U = matrix(U) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F +F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 59 "and respective substitutions, equation (4) assumes the \+ form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "'(B - lambda[B]*U)' * Y = (matrix(B) - root(B) * matrix(U)) * matrix(Y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"BG\"\"\"*&&%'lambdaG6#F&F'%\"UGF'!\"\"F'% \"YGF'*&,&-%'matrixG6#7%7%\"\"'\"\"$\"\"!7%F8F6F77%F8F8F6F'*&F6F'-F26# 7%7%F'F8F87%F8F'F87%F8F8F'F'F-F'-F26#7%7#\"\"*7#F8FGF'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "E valuation of this equation yields the " }{XPPEDIT 18 0 "zero" "6#%%ze roG" }{TEXT -1 24 " column matrix (vector)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "(matrix(B) - root(B) * matrix(U)) * matrix(Y) = ev alm((B - root(B) * U) &* Y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,& -%'matrixG6#7%7%\"\"'\"\"$\"\"!7%F-F+F,7%F-F-F+\"\"\"*&F+F0-F'6#7%7%F0 F-F-7%F-F0F-7%F-F-F0F0!\"\"F0-F'6#7%7#\"\"*7#F-F>F0-F'6#7%F>F>F>" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "which verifies that [" }{TEXT 343 1 "Y" }{TEXT -1 1 "]" } {TEXT 342 3 " = " }{TEXT -1 1 "[" }{TEXT 344 1 "S" }{TEXT -1 3 "] [" } {TEXT 345 1 "X" }{TEXT -1 26 "] is an eigenvector for [" }{TEXT 346 1 "B" }{TEXT -1 39 "] corresponding to the same eigenvalue." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 271 "" 0 "" {TEXT 347 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 427 3 "B. " }{TEXT 430 12 "Modal matrix" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "If the eigenvalues of a square matrix [" }{TEXT 348 1 "A" } {TEXT -1 7 "] are " }{TEXT 349 8 "distinct" }{TEXT -1 46 ", whether \+ real or complex, then a so-called " }{TEXT 350 5 "modal" }{TEXT -1 10 " matrix [" }{TEXT 351 1 "M" }{TEXT -1 19 "] associated with [" } {TEXT 352 1 "A" }{TEXT -1 47 "] is defined as a matrix of the same ord er as [" }{TEXT 353 1 "A" }{TEXT -1 73 "], having as its columns all t he (linearly independent) eigenvectors of [" }{TEXT 354 1 "A" }{TEXT -1 39 "] corresponding to the eigenvalues of [" }{TEXT 355 1 "A" } {TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Eigenvectors are " }{TEXT 356 20 "linear ly independent" }{TEXT -1 25 " if they correspond to " }{TEXT 357 8 "distinct" }{TEXT -1 14 " eigenvalues." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, cons ider a " }{TEXT 359 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 570 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 571 1 ")" }{TEXT -1 10 " \+ matrix [" }{TEXT 358 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "A := matrix(4, 4, [1, -2, 3, -4, -4, 1, - 2, 3, 3, -4, 1, -2, -2, 3, -4, 1]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&\"\"\"!\"#\"\"$!\"%7&F-F*F+F ,7&F,F-F*F+7&F+F,F-F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "and obtain the modal matrix [" } {TEXT 360 1 "M" }{TEXT -1 19 "] associated with [" }{TEXT 361 1 "A" } {TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 362 6 "Step 1" }{TEXT -1 27 ". Find the eigenval ues of [" }{TEXT 363 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = ch arroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG 6&!\"#\"#5^$F)\"\"#^$F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Extracting the four distinct eigen values " }{XPPEDIT 18 0 "lambda[i]" "6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "No_roots(A) \+ := nops([charroots(A)]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for i to No_roots(A) do ch_r[i](A) := charroots(A)[i] : print(l [i] = ch_r[i](A)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lamb daG6#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"# \"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"$^$!\"#\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"%^$!\"#F)" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 366 6 "Step 2" }{TEXT -1 10 ". Find the" }{TEXT 364 1 " " }{TEXT -1 62 "sequence of lists containing eigenvalues and eigenvectors of [" } {TEXT 365 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`roots&vectors(A)` := eigenvects(A) : roots_and_vectors(A) = `roots&vectors(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and _vectorsG6#%\"AG6&7%\"#5\"\"\"<#-%'vectorG6#7&!\"\"F+F1F+7%^$!\"#\"\"# F+<#-F.6#7&F1^#F1F+^#F+7%^$F4F4F+<#-F.6#7&F1F;F+F:7%F4F+<#-F.6#7&F+F+F +F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 572 6 "Step 3" }{TEXT -1 28 ". Extract the eigenvectors " } {XPPEDIT 18 0 "v[lambda[i]]" "6#&%\"vG6#&%'lambdaG6#%\"iG" }{TEXT -1 32 " corresponding to eigenvalues " }{XPPEDIT 18 0 "lambda[i]" "6#&% 'lambdaG6#%\"iG" }{TEXT -1 6 " of [" }{TEXT 573 1 "A" }{TEXT -1 2 "]: " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "As mentioned in Unit (21), the " }{TEXT 574 10 "eigenvect s" }{TEXT -1 269 " function returns an unordered sequence of the compo nent lists. To extract the eigenvectors so that each of them correspon ds to a proper eigenvalue, the following selection method is developed . The method works for matrices of any order, whose eigenvalues are di stinct." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 94 "for i to No_roots(A) do e[i] := charroots(A)[ i] : List[i] := `roots&vectors(A)`[i] : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "for j to No_roots(A) do for i to No_roots(A ) do if List[i][1] = e[j] then Lst[j] := `roots&vectors(A)`[i] : \+ fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "for \+ i to No_roots(A) do ch_v[i](A) := op(Lst[i][3]) : print(v[lambda[i] ] = eval(ch_v[i](A)), ` --> ` * lambda[i] = ch_r[i](A)) : od :" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lambdaG6#\"\"\"-%'vectorG6 #7&F*F*F*F*/*&%(~~-->~~GF*F'F*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ /&%\"vG6#&%'lambdaG6#\"\"#-%'vectorG6#7&!\"\"\"\"\"F/F0/*&%(~~-->~~GF0 F'F0\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lambdaG6#\"\"$ -%'vectorG6#7&!\"\"^#F/\"\"\"^#F1/*&%(~~-->~~GF1F'F1^$!\"#\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lambdaG6#\"\"%-%'vectorG6# 7&!\"\"^#\"\"\"F1^#F//*&%(~~-->~~GF1F'F1^$!\"#F7" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Conversion of the eigenvectors " }{XPPEDIT 18 0 "v[lambda[i]]" "6#&%\"vG6#&%'la mbdaG6#%\"iG" }{TEXT -1 24 " into column matrices " }{XPPEDIT 18 0 " X[lambda[i]]" "6#&%\"XG6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " yields" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "for i to No_roots(A) do X[ l[i]] := convert(ch_v[i](A), matrix) : print(X[l[i]] = matrix(X[l[i] ])) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6# \"\"\"-%'matrixG6#7&7#F*F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"XG6#&%'lambdaG6#\"\"#-%'matrixG6#7&7#!\"\"7#\"\"\"F/F1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"$-%'matrixG6#7&7#!\"\"7 #^#F07#\"\"\"7#^#F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lam bdaG6#\"\"%-%'matrixG6#7&7#!\"\"7#^#\"\"\"7#F37#^#F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 367 6 "Step 4 " }{TEXT -1 39 ". Construct the 'unique' modal matrix [" }{TEXT 368 1 "M" }{TEXT -1 19 "] associated with [" }{TEXT 369 1 "A" }{TEXT -1 2 "] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "i := 'i' : `seq(X[l [i]])` := seq(X[l[i]], i=1..coldim(A)) : M := augment(`seq(X[l[i]])` ) : M=matrix(M) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"MG-%'matrix G6#7&7&\"\"\"!\"\"F+F+7&F*F*^#F+^#F*7&F*F+F*F*7&F*F*F.F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "[ Refer to Unit (2) " }{TEXT 622 1 "H" }{TEXT -1 11 " where the " } {TEXT 575 7 "augment" }{TEXT -1 113 " function is used to create a rec tangular matrix from given column matrices. Also recall that its synon im is the " }{TEXT 627 6 "concat" }{TEXT -1 12 " function. ]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 280 "" 0 "" {TEXT 471 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 469 4 "N.B." }{TEXT -1 49 " In general, any per mutation of the columns of [" }{TEXT 470 1 "M" }{TEXT -1 69 "] will pr oduce another equally acceptable modal matrix. However, if [" }{TEXT 472 1 "M" }{TEXT -1 64 "] is to be used in computation of matrix funct ions (see Section " }{TEXT 473 1 "D" }{TEXT -1 44 " of this Unit), the ordering of columns in [" }{TEXT 474 1 "M" }{TEXT -1 3 "] " }{TEXT 19 4 "must" }{TEXT -1 59 " correspond to the unique sequence of the e igenvalues of [" }{TEXT 475 1 "A" }{TEXT -1 21 "] as returned by the \+ " }{TEXT 476 9 "eigenvals" }{TEXT -1 28 " function. This means that [ " }{TEXT 482 1 "M" }{TEXT -1 3 "] " }{TEXT 19 4 "must" }{TEXT -1 12 " be unique " }{TEXT 546 13 "in this sense" }{TEXT -1 33 ". However, s uch a uniqueness of [" }{TEXT 603 1 "M" }{TEXT -1 47 "] is not \"absol ute\" since the eigenvectors are " }{TEXT 604 3 "not" }{TEXT -1 27 " u nique for a given matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Consequently, the modal matrix th us created is referred to as 'unique' (enclosed in single quotes) in t his and subsequent Units." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "In this particular case, it has be en found that the eigenvectors corresponding to eigenvalues " } {XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" }{TEXT 577 1 "," } {TEXT -1 2 " " }{XPPEDIT 18 0 "lambda[3]" "6#&%'lambdaG6#\"\"$" } {TEXT 578 1 "," }{TEXT -1 7 " and " }{XPPEDIT 18 0 "lambda[4]" "6#&% 'lambdaG6#\"\"%" }{TEXT -1 134 " can be returned with their component s at different locations. Some of the obtained variants of the modal m atrix are presented below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 309 "M1 = matrix(4, 4, [-1, 1, I, -I, 1, 1, -1, -1, -1, 1, -I, I, 1, 1 , 1, 1]) ; M2 = matrix(4, 4, [1, 1, -I, I, -1, 1, 1, 1, 1, 1, I, -I, -1, 1, -1, -1]) ; M3 = matrix(4, 4, [-1, 1, 1, 1, 1, 1, I, -I, -1, \+ 1, -1, -1, 1, 1, -I, I]) ; M4 = matrix(4, 4, [-1, 1, -1, -1, 1, 1, - I, I, -1, 1, 1, 1, 1, 1, I, -I]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%#M1G-%'matrixG6#7&7&!\"\"\"\"\"^#F+^#F*7&F+F+F*F*7&F*F+F-F,7&F+F+F+F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M2G-%'matrixG6#7&7&\"\"\"F*^#! \"\"^#F*7&F,F*F*F*7&F*F*F-F+7&F,F*F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M3G-%'matrixG6#7&7&!\"\"\"\"\"F+F+7&F+F+^#F+^#F*7&F*F+F*F*7&F +F+F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M4G-%'matrixG6#7&7&!\"\" \"\"\"F*F*7&F+F+^#F*^#F+7&F*F+F+F+7&F+F+F.F-" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "However, permut ations of the elements in columns " }{TEXT 579 2 "1," }{TEXT -1 2 " \+ " }{TEXT 580 2 "3," }{TEXT -1 7 " and " }{TEXT 581 1 "4" }{TEXT -1 5 " do " }{TEXT 576 3 "not" }{TEXT -1 40 " affect computation of matr ix functions." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 272 "" 0 "" {TEXT 370 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 431 3 "C. " }{TEXT 432 27 "The Jordan form of a matrix" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 371 6 "Jordan" } {TEXT -1 8 " form [" }{TEXT 372 1 "J" }{TEXT -1 22 "] of a square mat rix [" }{TEXT 373 1 "A" }{TEXT -1 47 "] is defined as a matrix of the \+ same order as [" }{TEXT 374 1 "A" }{TEXT -1 50 "] whose diagonal eleme nts are the eigenvalues of [" }{TEXT 375 1 "A" }{TEXT -1 36 "] and who se other elements are all " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 377 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 582 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 583 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 376 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A := matr ix(3, 3, [3, 2, -1, 2, 3, -1, -1, -1, 4]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"$\"\"#!\"\"7 %F+F*F,7%F,F,\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and obtain its " }{TEXT 426 6 "Jordan" } {TEXT -1 7 " form." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 433 6 "Step 1" }{TEXT -1 27 ". Find the e igenvalues of [" }{TEXT 434 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots( A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG 6#%\"AG6%\"\"\"\"\"$\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Extracting the distinct eigenvalue s " }{XPPEDIT 18 0 "lambda[i]" "6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " y ields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "No_roots(A) := nop s([charroots(A)]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for \+ i to No_roots(A) do ch_r[i](A) := charroots(A)[i] : print(l[i] = ch _r[i](A)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\" \"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"$\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 438 6 "S tep 2" }{TEXT -1 17 ". Construct the " }{TEXT 435 6 "Jordan" }{TEXT -1 8 " form [" }{TEXT 437 1 "J" }{TEXT -1 6 "] of [" }{TEXT 436 1 "A " }{TEXT -1 2 "]:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "This may be done using either of the foll owing methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 584 8 "Method 1" }{TEXT -1 12 ". Using the " } {TEXT 585 4 "diag" }{TEXT -1 50 " function with manually entered diago nal elements:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "J := diag( ch_r[1](A), ch_r[2](A), ch_r[3](A)) : J = matrix(J) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"JG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+\"\"$F+7% F+F+\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 587 8 "Method 2" }{TEXT -1 19 ". Using the double " } {TEXT 586 3 "for" }{TEXT -1 24 "-loop construct and the " }{TEXT 588 8 "diagonal" }{TEXT -1 19 " indexing function:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "J_A := array(diagonal, 1..rowdim(J), 1..coldim (J)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "for i to rowdim( A) do for j to coldim(A) do if j = i then J_A[i,j] := ch_r[i](A) \+ fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "J := matrix(J_A) : J = matrix(J) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"JG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+\"\"$F+7%F+F+\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 279 "" 0 "" {TEXT 440 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 441 4 "N.B." }{TEXT -1 5 " In " }{TEXT 380 5 "Maple" } {TEXT -1 6 ", the " }{TEXT 381 6 "jordan" }{TEXT -1 33 " function may \+ be used to compute " }{TEXT 439 8 "directly" }{TEXT -1 6 " the " } {TEXT 378 6 "Jordan" }{TEXT -1 8 " form [" }{TEXT 379 1 "J" }{TEXT -1 37 "] of a matrix. For the above matrix [" }{TEXT 382 1 "A" }{TEXT -1 8 "], the " }{TEXT 383 6 "Jordan" }{TEXT -1 9 " form is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "J := jordan(A) : J = matri x(J) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"JG-%'matrixG6#7%7%\"\"\" \"\"!F+7%F+\"\"$F+7%F+F+\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 442 7 "WARNING" }{TEXT 443 1 ":" } {TEXT -1 50 " The order of matrix eigenvalues returned by the " } {TEXT 444 9 "eigenvals" }{TEXT -1 75 " function is always a unique, re producible sequence of the eigenvalues of [" }{TEXT 449 1 "A" }{TEXT -1 44 "] that may, for instance, be designated as " }{XPPEDIT 18 0 "l ambda[1]" "6#&%'lambdaG6#\"\"\"" }{TEXT 445 3 ", " }{XPPEDIT 18 0 "la mbda[2]" "6#&%'lambdaG6#\"\"#" }{TEXT 446 3 ", " }{XPPEDIT 18 0 "lamb da[3]" "6#&%'lambdaG6#\"\"$" }{TEXT 447 7 ", ..., " }{XPPEDIT 18 0 "la mbda[n]" "6#&%'lambdaG6#%\"nG" }{TEXT 448 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 450 6 "jordan" }{TEXT -1 23 " function returns the " }{TEXT 451 6 "Jo rdan" }{TEXT -1 8 " form [" }{TEXT 453 1 "J" }{TEXT -1 6 "] of [" } {TEXT 452 1 "A" }{TEXT -1 120 "] that is unique up to permutations of \+ the diagonal entries. In other words, the locations of the diagonal el ements in [" }{TEXT 468 1 "J" }{TEXT -1 6 "] are " }{TEXT 467 3 "not" }{TEXT -1 47 " unique. This implies that the eigenvalues of [" }{TEXT 454 1 "A" }{TEXT -1 71 "] may, occasionally, appear at different locat ions of the diagonal of [" }{TEXT 455 1 "J" }{TEXT -1 36 "], which, in general, is acceptable." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Where it is essential that locatio ns of the diagonal elements of [" }{TEXT 460 1 "J" }{TEXT -1 67 "] cor respond to the unique ordering of eigenvalues returned by the " } {TEXT 459 9 "eigenvals" }{TEXT -1 16 " function, the " }{TEXT 456 6 " Jordan" }{TEXT -1 8 " form [" }{TEXT 458 1 "J" }{TEXT -1 6 "] of [" } {TEXT 457 1 "A" }{TEXT -1 161 "] should be constructed using Method 1 \+ or 2. This is an imperative in computation of matrix functions using s imilarity of matrices \226 refer to Units (23) to (25)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 273 "" 0 "" {TEXT 384 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 501 4 "N.B." }{TEXT -1 23 " Every square matrix [" }{TEXT 385 1 "A" }{TEXT -1 6 "] is " }{TEXT 386 7 "similar" }{TEXT -1 21 " \+ to the associated " }{TEXT 387 6 "Jordan" }{TEXT -1 8 " form [" } {TEXT 510 1 "J" }{TEXT -1 39 "] of the matrix. This implies that if [ " }{TEXT 388 1 "M" }{TEXT -1 25 "] is a modal matrix for [" }{TEXT 389 1 "A" }{TEXT -1 7 "], then" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 1 "[" }{TEXT 391 1 "A" }{TEXT -1 1 "]" }{TEXT 390 3 " = " }{TEXT -1 1 "[" }{TEXT 393 1 "M" }{TEXT -1 3 "] [" }{TEXT 395 1 "J" }{TEXT -1 2 "] " }{TEXT 394 3 "Inv" }{TEXT -1 1 "[" }{TEXT 392 1 "M" }{TEXT -1 1 "] " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "or, equivalently," }}}{EXCHG {PARA 284 "" 0 "" {TEXT -1 1 "[" }{TEXT 529 1 "A" }{TEXT -1 3 "] [" }{TEXT 532 1 "M" }{TEXT -1 1 "]" }{TEXT 528 3 " = " }{TEXT -1 1 "[" }{TEXT 530 1 "M" }{TEXT -1 3 "] [" }{TEXT 531 1 "J" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Exemplarily, consider t he same matrix [" }{TEXT 396 1 "A" }{TEXT -1 17 "] as before, i.e." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"$\"\"#!\"\"7%F+F*F ,7%F,F,\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "(1) The 'unique' modal matrix [" }{TEXT 589 1 " M" }{TEXT -1 7 "] for [" }{TEXT 590 1 "A" }{TEXT -1 5 "] is:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "`roots&vectors(A)` := eigenv ects(A) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "for i to No_ro ots(A) do e[i] := charroots(A)[i] : List[i] := `roots&vectors(A)`[i ] : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "for j to No_ roots(A) do for i to No_roots(A) do if List[i][1] = e[j] then Lst [j] := `roots&vectors(A)`[i] : fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "for i to No_roots(A) do ch_v[i](A) := o p(Lst[i][3]) : X[l[i]] := convert(ch_v[i](A), matrix) : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "i := 'i' : `seq(X[l[i]])` := seq(X[l[i]], i=1..coldim(A)) : M := augment(`seq(X[l[i]])`) : \+ M=matrix(M) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"MG-%'matrixG6#7%7 %!\"\"\"\"\"F*7%F+F+F*7%\"\"!\"\"#F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(2) With the unique " }{TEXT 591 6 "Jordan" }{TEXT -1 8 " form [" }{TEXT 592 1 "J" }{TEXT -1 6 "] of [" }{TEXT 593 1 "A" }{TEXT -1 45 "] computed earlier (see M ethod 2 above), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "J : = matrix(J_A) : J = matrix(J) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"JG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+\"\"$F+7%F+F+\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "t he matrix product [" }{TEXT 606 1 "M" }{TEXT -1 3 "] [" }{TEXT 608 1 "J" }{TEXT -1 2 "] " }{TEXT 607 3 "Inv" }{TEXT -1 1 "[" }{TEXT 605 1 " M" }{TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "`M J Inv(M)` := evalm(M &* J &* M^(-1)) : M*J*Inv(M ) = matrix(`M J Inv(M)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"MG \"\"\"%\"JGF&-%$InvG6#F%F&-%'matrixG6#7%7%\"\"$\"\"#!\"\"7%F1F0F27%F2F 2\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "which is equal to matrix [" }{TEXT 397 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 26 "(3a) The matrix product [" }{TEXT 533 1 "A" } {TEXT -1 3 "] [" }{TEXT 534 1 "M" }{TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "`AM` := evalm(A &* M) : \+ A*M = matrix(`AM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\" \"%\"MGF&-%'matrixG6#7%7%!\"\"\"\"$!\"'7%F&F.F/7%\"\"!\"\"'F3" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "(3b) The matrix product [" }{TEXT 535 1 "M" }{TEXT -1 3 "] [" }{TEXT 536 1 "J" }{TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`MJ` := evalm(M &* J) : `M J` = matrix( `MJ`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$M~JG-%'matrixG6#7%7%!\" \"\"\"$!\"'7%\"\"\"F+F,7%\"\"!\"\"'F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Both product matr ices of (3a) and (3b) are equal, which verifies that matrix [" }{TEXT 564 1 "A" }{TEXT -1 17 "] is similar to [" }{TEXT 565 1 "J" }{TEXT -1 52 "]. This verification may be done directly using the " }{TEXT 566 9 "issimilar" }{TEXT -1 15 " function, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "issimilar(A, J) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 653 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 18 " returned by the " }{TEXT 567 9 "issimilar" }{TEXT -1 28 " function verifies the fact." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 282 "" 0 "" {TEXT 500 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 509 4 "N.B." }{TEXT -1 7 " The " }{TEXT 502 6 "Jordan" }{TEXT -1 8 " form [" }{TEXT 511 1 "J" }{TEXT -1 22 "] of a \+ square matrix [" }{TEXT 512 1 "A" }{TEXT -1 6 "] is " }{TEXT 514 7 "s imilar" }{TEXT -1 6 " to [" }{TEXT 513 1 "A" }{TEXT -1 25 "]. This im plies that if [" }{TEXT 503 1 "M" }{TEXT -1 25 "] is a modal matrix fo r [" }{TEXT 504 1 "A" }{TEXT -1 7 "], then" }}}{EXCHG {PARA 283 "" 0 " " {TEXT -1 1 "[" }{TEXT 506 1 "J" }{TEXT -1 1 "]" }{TEXT 505 6 " = Inv " }{TEXT -1 1 "[" }{TEXT 515 1 "M" }{TEXT -1 3 "] [" }{TEXT 507 1 "A" }{TEXT -1 3 "] [" }{TEXT 508 1 "M" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Exemplaril y, using the same matrix [" }{TEXT 518 1 "A" }{TEXT -1 53 "] as before together with its 'unique' modal matrix [" }{TEXT 519 1 "M" }{TEXT -1 18 "] and the unique " }{TEXT 520 6 "Jordan" }{TEXT -1 8 " form [ " }{TEXT 521 1 "J" }{TEXT -1 6 "] of [" }{TEXT 522 1 "A" }{TEXT -1 37 "], evaluation of the matrix product " }{TEXT 524 3 "Inv" }{TEXT -1 1 "[" }{TEXT 523 1 "M" }{TEXT -1 3 "] [" }{TEXT 516 1 "A" }{TEXT -1 3 "] [" }{TEXT 517 1 "M" }{TEXT -1 20 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "`Inv(M) A M` := evalm(M^(-1) &* A & * M) : Inv(M)*A*M = matrix(`Inv(M) A M`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%$InvG6#%\"MG\"\"\"%\"AGF)F(F)-%'matrixG6#7%7%F)\" \"!F07%F0\"\"$F07%F0F0\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "which is equal to the unique " }{TEXT 525 6 "Jordan" }{TEXT -1 8 " form [" }{TEXT 526 1 "J" }{TEXT -1 6 "] of [" }{TEXT 527 1 "A" }{TEXT -1 90 "]. This verifies the abov e relationship. This verification may be done directly using the " } {TEXT 568 9 "issimilar" }{TEXT -1 15 " function, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "issimilar(J, A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 654 7 "Boolean" } {TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 18 " return ed by the " }{TEXT 569 9 "issimilar" }{TEXT -1 28 " function verifies \+ the fact." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 275 "" 0 "" {TEXT 398 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 461 3 "D. " }{TEXT 462 32 "Applicati on of matrix similarity" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "The foregoing considerations are \+ useful in computing functions of square matrices. For this purpose, it is convenient to divide matrices into two classes as follows." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(1) Diagonal matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "If [" }{TEXT 399 1 "A" }{TEXT -1 150 "] is a diagonal matrix, then a function of the matrix is also \+ a diagonal matrix of the same order whose elements are the function of the elements of [" }{TEXT 400 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "For exa mple, if [" }{TEXT 463 1 "A" }{TEXT -1 29 "] is given as the following " }{TEXT 464 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 465 3 " \327 \+ " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 466 1 ")" }{TEXT -1 15 " scalar matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "l := 'l' : l : = lambda :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A := diag(l[1 ], l[2], l[3]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"AG-%'matrixG6#7%7%&%'lambdaG6#\"\"\"\"\"!F.7%F.&F+6#\"\"#F.7%F.F.& F+6#\"\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 17 "then a function " }{TEXT 547 1 "f" }{TEXT -1 82 " of the matrix may be obtained using either of the following alternat ive methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 550 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 549 5 "array" }{TEXT -1 28 " function together with the " }{TEXT 548 8 "diagonal" }{TEXT -1 19 " indexing function:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 108 "f(A) := array(diagonal, 1..3, 1..3, [(1,1)=f( A[1,1]), (2,2)=f(A[2,2]), (3,3)=f(A[3,3])]) : 'f(A)' = f(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"AG-%'matrixG6#7%7%-F%6#&%' lambdaG6#\"\"\"\"\"!F37%F3-F%6#&F06#\"\"#F37%F3F3-F%6#&F06#\"\"$" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 594 4 "N.B." }{TEXT -1 6 " The " }{TEXT 595 4 "diag" }{TEXT -1 10 " f unction " }{TEXT 596 6 "cannot" }{TEXT -1 59 " be used to construct a \+ diagonal matrix whose entries are " }{TEXT 597 9 "functions" }{TEXT -1 76 " of scalars because this function works only if the diagonal e lements are " }{TEXT 598 7 "scalars" }{TEXT -1 49 " in numerical or \+ symbolic form. That is why the " }{TEXT 600 5 "array" }{TEXT -1 28 " f unction together with the " }{TEXT 599 8 "diagonal" }{TEXT -1 20 " ind exing function " }{TEXT 19 4 "must" }{TEXT -1 35 " be used if this m ethod is chosen." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "If large diagonal matrices are involved, \+ the method proposed below will be more efficient." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 551 8 "Method 2" }{TEXT -1 19 ". Using the double " }{TEXT 552 3 "for" }{TEXT -1 24 "-l oop construct and the " }{TEXT 601 8 "diagonal" }{TEXT -1 19 " indexin g function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a := array(d iagonal, 1..rowdim(A), 1..coldim(A)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "for i to rowdim(A) do for j to coldim(A) do if j \+ = i then a[i,j] := f(A[i,j]) fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f(A) := matrix(a) : 'f(A)' = f(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"AG-%'matrixG6#7%7%-F%6#&%' lambdaG6#\"\"\"\"\"!F37%F3-F%6#&F06#\"\"#F37%F3F3-F%6#&F06#\"\"$" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 553 4 "N.B." }{TEXT -1 54 " The construction of this method is such t hat it has " }{TEXT 555 2 "no" }{TEXT -1 12 " effect on [" }{TEXT 554 1 "A" }{TEXT -1 129 "] in view of a possible need for using the matrix in further computations. For instance, this is the case where verific ation of " }{TEXT 558 2 "f(" }{TEXT -1 1 "[" }{TEXT 557 1 "A" }{TEXT -1 1 "]" }{TEXT 559 1 ")" }{TEXT -1 9 " using " }{TEXT 556 10 "Macla urin\222" }{TEXT -1 53 "s series is performed \226 refer to Units (23 ) to (25)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(2) Non-diagonal matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "If a non-dia gonal matrix [" }{TEXT 401 1 "A" }{TEXT -1 6 "] is " }{TEXT 402 7 "si milar" }{TEXT -1 21 " to the associated " }{TEXT 403 6 "Jordan" } {TEXT -1 29 " form of the matrix, i.e. if" }}}{EXCHG {PARA 277 "" 0 " " {TEXT -1 1 "[" }{TEXT 405 1 "A" }{TEXT -1 1 "]" }{TEXT 404 3 " = " } {TEXT -1 1 "[" }{TEXT 407 1 "M" }{TEXT -1 3 "] [" }{TEXT 409 1 "J" } {TEXT -1 2 "] " }{TEXT 408 3 "Inv" }{TEXT -1 1 "[" }{TEXT 406 1 "M" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "then a function of [" }{TEXT 410 1 "A" } {TEXT -1 13 "] is given by" }}}{EXCHG {PARA 278 "" 0 "" {TEXT 416 2 "f (" }{TEXT -1 1 "[" }{TEXT 411 1 "A" }{TEXT -1 1 "]" }{TEXT 417 4 ") = \+ " }{TEXT -1 1 "[" }{TEXT 413 1 "M" }{TEXT -1 2 "] " }{TEXT 418 2 "f(" }{TEXT -1 1 "[" }{TEXT 415 1 "J" }{TEXT -1 1 "]" }{TEXT 419 1 ")" } {TEXT -1 1 " " }{TEXT 414 3 "Inv" }{TEXT -1 1 "[" }{TEXT 412 1 "M" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where [" }{TEXT 420 1 "M" }{TEXT -1 39 "] \+ is the modal matrix associated with [" }{TEXT 421 1 "A" }{TEXT -1 2 "] ." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The formula requires that two necessary conditions are sa tisfied, viz." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "(a) The ordering of eigenvector columns i n [" }{TEXT 477 1 "M" }{TEXT -1 3 "] " }{TEXT 19 4 "must" }{TEXT -1 59 " correspond to the unique sequence of the eigenvalues of [" } {TEXT 478 1 "A" }{TEXT -1 21 "] as returned by the " }{TEXT 479 9 "eig envals" }{TEXT -1 10 " function." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "(b) The locations of the d iagonal elements of [" }{TEXT 481 1 "J" }{TEXT -1 3 "] " }{TEXT 19 4 "must" }{TEXT -1 67 " correspond to the unique ordering of eigenvalue s returned by the " }{TEXT 480 9 "eigenvals" }{TEXT -1 10 " function. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Both conditions may be summarised as follows:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 " \"The diagonal elements of [" }{TEXT 643 1 "J" }{TEXT -1 59 "] are the eigenvalues associated with the eigenvectors of [" }{TEXT 644 1 "A" } {TEXT -1 61 "], in the same order as their corresponding eigenvectors \+ in [" }{TEXT 645 1 "M" }{TEXT -1 3 "].\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 286 "" 0 "" {TEXT 602 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (23) for \"" }{TEXT 423 30 "Functions of consta nt matrices" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 276 "" 0 "" {TEXT 422 67 "------------------------------------------------------------------ -" }}}}{MARK "6 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }