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1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 286 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 287 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 288 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 273 "" 0 "" {TEXT 452 39 "MATRICES AND MATRIX OPE RATIONS: Unit 15" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 274 "" 0 "" {TEXT 454 23 "Dr. Wlodzislaw Kostecki" }}{PARA 275 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 276 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 277 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 278 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT 453 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 280 "" 0 "" {TEXT 455 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT 456 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 267 4 "(15)" }{TEXT 315 1 " " }{TEXT 314 34 "Integer exponenti ation of matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 546 10 "OBJECTIVES" }{TEXT 547 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 548 1 " \225" }{TEXT -1 71 " To define the operation of integer exponentiatio n of square matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 549 1 "\225" } {TEXT -1 66 " To provide alternative methods of exponentiation operat ion with " }{TEXT 550 5 "Maple" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 551 1 "\225" }{TEXT -1 42 " To provide definitions and exa mples of " }{TEXT 552 9 "nilpotent" }{TEXT -1 7 " and " }{TEXT 553 10 "idempotent" }{TEXT -1 43 " matrices and illustrate their properti es." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 555 1 "\225" }{TEXT -1 79 " To s how the effect of exponentiation operation on specific types of matric es." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 577 1 "\225" }{TEXT -1 72 " To s pecify and illustrate some properties of exponentiation operation." }} }{EXCHG {PARA 0 "" 0 "" {TEXT 564 1 "\225" }{TEXT -1 63 " To stress a nd show that a square matrix raised to the power " }{XPPMATH 20 "6#%% zeroG" }{TEXT -1 12 " yields in " }{TEXT 565 5 "Maple" }{TEXT -1 19 " the scalar value " }{XPPMATH 20 "6#%$oneG" }{TEXT -1 14 ", and not \+ a " }{TEXT 566 4 "zero" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "restar t : with(linalg, definite, det, diag, eigenvals, inverse, multiply, \+ transpose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Positive- and negative-integer exponentiation ( integer power) of a matrix is defined " }{TEXT 257 4 "only" }{TEXT -1 138 " for square matrices. Integer exponentiation of a matrix is multi ple multiplication of the matrix with itself. Therefore, to satisfy th e " }{TEXT 538 34 "multiplication conformability rule" }{TEXT -1 42 " , this operation requires that a matrix " }{TEXT 19 4 "must" }{TEXT -1 12 " be square." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 3 "A. " }{TEXT 276 34 "Positive-inte ger power of a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "The positive-integer power of a sq uare matrix of order " }{TEXT 374 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 375 3 " \327 " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 376 1 ")" } {TEXT -1 6 " or " }{TEXT 377 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT 378 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 379 1 ")" } {TEXT -1 181 " is a matrix of the same order. The resultant is a prod uct matrix that is obtained by multiple multiplication of a given matr ix with itself. For example, the square of the matrix [" }{TEXT 258 1 "A" }{TEXT -1 28 "] is defined by the relation" }}}{EXCHG {PARA 258 " " 0 "" {TEXT -1 1 "[" }{TEXT 269 1 "A" }{TEXT -1 1 "]" }{TEXT 270 1 "^ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 380 3 " = " }{TEXT -1 1 "[" } {TEXT 259 1 "A" }{TEXT -1 3 "] [" }{TEXT 260 1 "A" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 381 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 382 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 383 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 261 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(2, 2 , [a[11], a[12], a[21], a[22]]) : A = matrix(A) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\"#77$&F+6#\"# @&F+6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The square of the matrix, [" }{TEXT 273 1 "A" } {TEXT -1 1 "]" }{TEXT 274 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 69 ", may be obtained using either of the following alternative me thods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 363 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 364 5 " evalm" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`A^2` := evalm(A^2) : A^2 = matrix(`A^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"#\"\"\"-%'matrixG6#7$7$,&*$)&%\"aG6# \"#6F'F(F(*&&F26#\"#7F(&F26#\"#@F(F(,&*&F1F(F6F(F(*&F6F(&F26#\"#AF(F(7 $,&*&F9F(F1F(F(*&F?F(F9F(F(,&F5F(*$)F?F'F(F(" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 365 8 "Method 2" } {TEXT -1 12 ". Using the " }{TEXT 366 8 "multiply" }{TEXT -1 10 " func tion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`A^2` := multiply( A, A) : A^2 = matrix(`A^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$ )%\"AG\"\"#\"\"\"-%'matrixG6#7$7$,&*$)&%\"aG6#\"#6F'F(F(*&&F26#\"#7F(& F26#\"#@F(F(,&*&F1F(F6F(F(*&F6F(&F26#\"#AF(F(7$,&*&F9F(F1F(F(*&F?F(F9F (F(,&F5F(*$)F?F'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "As a numerical example, compute th e square of a " }{TEXT 384 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 385 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 386 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 262 1 "A" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := matrix(2, 2, [1, 2, 2, \+ -4]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'m atrixG6#7$7$\"\"\"\"\"#7$F+!\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The square of [" }{TEXT 275 1 "A" }{TEXT -1 20 "] is the following " }{TEXT 387 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 388 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 389 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`A^2` := evalm(A^2) : A^2 = matrix(`A^2`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"#\"\"\"-%'matrixG6#7$7$\" \"&!\"'7$F/\"#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`A^2` := multiply(A, A) : A^2 = matrix(`A^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"#\"\"\"-%'matrixG6#7$7$\"\"&!\"'7$ F/\"#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 288 " " 0 "" {TEXT 567 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 568 4 "N.B." }{TEXT -1 20 " The above ma trix [" }{TEXT 569 1 "A" }{TEXT -1 15 "] is a unique " }{TEXT 570 11 "square root" }{TEXT -1 14 " of matrix [" }{TEXT 571 1 "A" }{TEXT -1 1 "]" }{TEXT 572 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 25 " \+ since the latter is a " }{TEXT 573 17 "positive definite" }{TEXT -1 29 " matrix as verified by the " }{TEXT 579 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 27 " returned by the f unction " }{TEXT 574 8 "definite" }{TEXT -1 6 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "definite(`A^2`, 'positive_def') ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "[ For the functio n " }{TEXT 575 11 "square root" }{TEXT -1 32 " of a matrix, refer to Section " }{TEXT 576 1 "B" }{TEXT -1 16 " of Unit (23). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT 277 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 4 "N.B." }{TEXT -1 72 " If a square matrix vanishes up on being raised to some positive power " }{XPPEDIT 18 0 "p" "6#%\"pG " }{TEXT -1 34 ", then the matrix is said to be " }{TEXT 278 9 "nilp otent" }{TEXT -1 12 " of index " }{XPPEDIT 18 0 "p" "6#%\"pG" } {TEXT -1 88 ". This implies that raising such a matrix to any integer power greater than the index " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 25 " will also result in a " }{XPPEDIT 18 0 "zero" "6#%%zeroG" } {TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 390 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 391 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 392 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 281 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A := matrix(3, 3, [1, 5, -2, 1, 2, -1, 3, 6, -3]) \+ : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrix G6#7%7%\"\"\"\"\"&!\"#7%F*\"\"#!\"\"7%\"\"$\"\"'!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "and the index " }{XPPEDIT 18 0 "p = 3" "6#/%\"pG\"\"$" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The matrix [" }{TEXT 282 1 "A" }{TEXT -1 35 "] raised to this p ower yields the " }{TEXT 393 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 394 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 395 1 ")" } {TEXT -1 2 " " }{TEXT 580 4 "zero" }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`A^3` := evalm(A^3) : A^3 = matri x(`A^3`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"$\"\"\"-%'m atrixG6#7%7%\"\"!F.F.F-F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "or, the matrix [" }{TEXT 283 1 "A " }{TEXT -1 41 "] has vanished upon raising it to power " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Notice that the determinan t of a " }{TEXT 541 9 "nilpotent" }{TEXT -1 13 " matrix is " } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Notice that \+ eigenvalues of a " }{TEXT 542 9 "nilpotent" }{TEXT -1 18 " matrix ar e all " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A ) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6 #%\"AG6%\"\"!F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "[ For " }{TEXT 554 11 "eigenvalues" } {TEXT -1 36 " of matrices, refer to Unit (21). ]" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 265 "" 0 "" {TEXT 280 5 "* * *" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 4 "N.B." }{TEXT -1 97 " If a square matrix is unchanged und er multiplication by itself, then the matrix is said to be " }{TEXT 285 10 "idempotent" }{TEXT -1 78 ". This implies that raising such a \+ matrix to any integer power greater than " }{XPPEDIT 18 0 "2" "6#\"\" #" }{TEXT -1 34 " will also not change the matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exempla rily, consider a " }{TEXT 396 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 397 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 398 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 286 1 "A" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A := matrix(3, 3, [2, -2, - 4, -1, 3, 4, 1, -2, -3]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#!\"#!\"%7%!\"\"\"\"$\"\"%7% \"\"\"F+!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The square of [" }{TEXT 287 1 "A" }{TEXT -1 20 "] is the same matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`A ^2` := evalm(A^2) : A^2 = matrix(`A^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"#\"\"\"-%'matrixG6#7%7%F'!\"#!\"%7%!\"\"\" \"$\"\"%7%F(F.!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Notice that the determinant of an " } {TEXT 543 10 "idempotent" }{TEXT -1 13 " matrix is " }{XPPMATH 20 "6 #%%zeroG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 288 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 4 "N .B." }{TEXT -1 57 " If a diagonal matrix is raised to some positive p ower " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 50 ", then its element s are raised to the same power." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " } {TEXT 399 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 400 3 " \327 " } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 401 1 ")" }{TEXT -1 19 " diagonal \+ matrix [" }{TEXT 290 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A := diag(a[11], 0, a[33], a[44]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7 &&%\"aG6#\"#6\"\"!F.F.7&F.F.F.F.7&F.F.&F+6#\"#LF.7&F.F.F.&F+6#\"#W" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "and the power " }{XPPEDIT 18 0 "p=5" "6#/%\"pG\"\"&" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The matrix [" }{TEXT 291 1 "A" }{TEXT -1 60 "] raise d to this power yields the following diagonal matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`A^5` := evalm(A^5) : A^5 = matri x(`A^5`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"&\"\"\"-%'m atrixG6#7&7&*$)&%\"aG6#\"#6F'F(\"\"!F4F47&F4F4F4F47&F4F4*$)&F16#\"#LF' F(F47&F4F4F4*$)&F16#\"#WF'F(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 292 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 4 "N.B." } {TEXT -1 80 " The transpose of a square matrix raised to some (positi ve or negative) power " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 48 " \+ is equal to the transpose raised to this power" }}}{EXCHG {PARA 268 " " 0 "" {TEXT 296 7 "Transp(" }{TEXT -1 1 "[" }{TEXT 294 1 "A" }{TEXT -1 1 "]" }{TEXT 298 1 "^" }{TEXT 299 1 "p" }{TEXT 372 1 ")" }{TEXT -1 1 " " }{TEXT 295 1 "=" }{TEXT -1 1 " " }{TEXT 297 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 300 1 "A" }{TEXT -1 1 "]" }{TEXT 373 2 ")^" }{TEXT 301 1 "p" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 402 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 403 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 404 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 302 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " A := matrix(3, 3, [2, -1, 4, 1, 3, 3, -1, 2, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#!\"\"\"\" %7%\"\"\"\"\"$F/7%F+F*\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "and the power " }{XPPEDIT 18 0 "p=3" "6#/%\"pG\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "(a) The transpose of th e matrix raised to this power, " }{TEXT 304 7 "Transp(" }{TEXT -1 1 " [" }{TEXT 303 1 "A" }{TEXT -1 1 "]" }{TEXT 305 1 "^" }{XPPEDIT 18 0 "3 " "6#\"\"$" }{TEXT 405 1 ")" }{TEXT -1 21 ", is the following " } {TEXT 406 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 407 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 408 1 ")" }{TEXT -1 9 " matrix:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "`transp(A^3)` := transpose (evalm(A^3)) : Transp(A^3) = matrix(`transp(A^3)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#*$)%\"AG\"\"$\"\"\"-%'matrixG6#7%7%! \"%\"\"&F27%\"#?\"#m\"#D7%F2\"#]\"#@" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "(b) The transpose raise d to this power, " }{TEXT 306 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 307 1 "A" }{TEXT -1 1 "]" }{TEXT 412 2 ")^" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT -1 21 ", is the following " }{TEXT 409 1 "(" }{XPPEDIT 18 0 "3 " "6#\"\"$" }{TEXT 410 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 411 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`(transp(A))^3` := evalm((transpose(A))^3) : [Trans p(A)]^3 = matrix(`(transp(A))^3`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*$)7#-%'TranspG6#%\"AG\"\"$\"\"\"-%'matrixG6#7%7%!\"%\"\"&F37%\"#?\" #m\"#D7%F3\"#]\"#@" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Both resultant matrices are equal." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 316 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 4 "N.B." }{TEXT -1 27 " If two square matri ces, [" }{TEXT 318 1 "A" }{TEXT -1 7 "] and [" }{TEXT 319 1 "B" } {TEXT -1 8 "], are " }{TEXT 320 9 "commuting" }{TEXT -1 21 " matrice s, then and " }{TEXT 362 4 "only" }{TEXT -1 5 " then" }}}{EXCHG {PARA 271 "" 0 "" {TEXT 413 1 "(" }{TEXT -1 1 "[" }{TEXT 321 1 "A" }{TEXT -1 1 "]" }{TEXT 414 3 " + " }{TEXT -1 1 "[" }{TEXT 322 1 "B" }{TEXT -1 2 "])" }{TEXT 323 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 415 3 " = " }{TEXT -1 1 "[" }{TEXT 324 1 "A" }{TEXT -1 1 "]" }{TEXT 325 1 "^ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 416 3 " + " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT -1 1 "[" }{TEXT 326 1 "A" }{TEXT -1 2 "][" }{TEXT 327 1 "B" }{TEXT -1 1 "]" }{TEXT 417 3 " + " }{TEXT -1 1 "[" }{TEXT 328 1 "B" }{TEXT -1 1 "]" }{TEXT 329 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "otherwise" }}}{EXCHG {PARA 272 "" 0 "" {TEXT 418 1 "(" }{TEXT -1 1 "[" }{TEXT 330 1 "A" }{TEXT -1 1 "]" }{TEXT 419 3 " + " }{TEXT -1 1 "[" }{TEXT 331 1 "B" }{TEXT -1 2 "])" }{TEXT 332 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 420 3 " = " }{TEXT -1 1 "[" }{TEXT 333 1 "A " }{TEXT -1 1 "]" }{TEXT 334 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 421 3 " + " }{TEXT -1 1 "[" }{TEXT 335 1 "A" }{TEXT -1 2 "][" } {TEXT 336 1 "B" }{TEXT -1 1 "]" }{TEXT 422 3 " + " }{TEXT -1 1 "[" } {TEXT 337 1 "B" }{TEXT -1 2 "][" }{TEXT 338 1 "A" }{TEXT -1 1 "]" } {TEXT 423 3 " + " }{TEXT -1 1 "[" }{TEXT 339 1 "B" }{TEXT -1 1 "]" } {TEXT 340 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 341 27 "Exemplarily, c onsider two " }{TEXT 424 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 344 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 345 1 ")" }{TEXT 425 22 " commuting matrices [" }{TEXT 342 1 "A" }{TEXT 346 7 "] and [ " }{TEXT 343 1 "B" }{TEXT 347 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "A := matrix(2, 2, [3, 4, 2, 3]) : B := matrix (2, 2, [3, -4, -2, 3]) : A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"$\"\"%7$\"\"#F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\"\"$!\"%7$!\"# F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "(a) The sum of the two matrices squared, " }{TEXT 426 1 "(" }{TEXT -1 1 "[" }{TEXT 351 1 "A" }{TEXT -1 1 "]" }{TEXT 427 3 " + \+ " }{TEXT -1 1 "[" }{TEXT 352 1 "B" }{TEXT -1 1 "]" }{TEXT 428 2 ")^" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 20 ", is the following " } {TEXT 348 1 " " }{TEXT 429 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 349 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 350 1 ")" }{TEXT 430 16 " scalar matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`(A+B)^2` := evalm((A+B)^2) : (A + B)^`2` = matrix(`(A+B)^2`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/),&%\"AG\"\"\"%\"BGF'%\"2G-%'matrixG6 #7$7$\"#O\"\"!7$F0F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "(b) The sum [" }{TEXT 353 1 "A" } {TEXT -1 1 "]" }{TEXT 354 1 "^" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 431 3 " + " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 1 "[" }{TEXT 355 1 "A" }{TEXT -1 2 "][" }{TEXT 356 1 "B" }{TEXT -1 1 "]" }{TEXT 432 3 " + " }{TEXT -1 1 "[" }{TEXT 357 1 "B" }{TEXT -1 1 "]" }{TEXT 358 1 "^" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 19 " is the following " }{TEXT 359 1 " " }{TEXT 433 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 360 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 361 1 ")" }{TEXT 434 16 " \+ scalar matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`A^2+2AB +B^2` := evalm(A^2 + 2*A &* B + B^2) : A^2+2*A*B+B^2 = matrix(`A^2+2 AB+B^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"AG\"\"#\"\"\"F) *(F(F)F'F)%\"BGF)F)*$)F+F(F)F)-%'matrixG6#7$7$\"#O\"\"!7$F4F3" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The resultant matrices of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 308 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 479 4 "N.B." }{TEXT -1 67 " In general, the integer power \+ of the product of square matrices [" }{TEXT 488 1 "A" }{TEXT -1 7 "] a nd [" }{TEXT 489 1 "B" }{TEXT -1 23 "] of the same order is " }{TEXT 495 3 "not" }{TEXT -1 63 " equal to the product of either matrix raise d to the same power" }}}{EXCHG {PARA 285 "" 0 "" {TEXT 484 1 "(" } {TEXT -1 1 "[" }{TEXT 480 1 "A" }{TEXT -1 3 "] [" }{TEXT 481 1 "B" } {TEXT -1 1 "]" }{TEXT 485 3 ")^n" }{TEXT -1 2 " " }{TEXT 19 15 "is no t equal to" }{TEXT -1 3 " [" }{TEXT 482 1 "A" }{TEXT -1 1 "]" }{TEXT 486 3 "^n " }{TEXT -1 1 "[" }{TEXT 483 1 "B" }{TEXT -1 1 "]" }{TEXT 487 2 "^n" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "unless the matrices are " }{TEXT 496 9 "commuting " }{TEXT -1 11 " matrices " }{TEXT 544 1 "(" }{TEXT -1 12 "for which \+ [" }{TEXT 490 1 "A" }{TEXT -1 3 "] [" }{TEXT 491 1 "B" }{TEXT -1 1 "] " }{TEXT 494 3 " = " }{TEXT -1 1 "[" }{TEXT 492 1 "B" }{TEXT -1 3 "] [ " }{TEXT 493 1 "A" }{TEXT -1 1 "]" }{TEXT 545 1 ")" }{TEXT -1 6 " or \+ " }{TEXT 497 8 "diagonal" }{TEXT -1 11 " matrices." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "(1) Exe mplarily, let the natural number " }{XPPEDIT 18 0 "n= 2" "6#/%\"nG\" \"#" }{TEXT -1 7 " and " }{TEXT 499 1 "(" }{XPPEDIT 18 0 "3" "6#\"\" $" }{TEXT 500 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 501 1 ") " }{TEXT -1 12 " matrices [" }{TEXT 498 1 "A" }{TEXT -1 7 "] and [" } {TEXT 502 1 "B" }{TEXT -1 13 "] be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "A := matrix(3, 3, [2, 3, 1, 2, -2, -2, -1, 2, 1]) : B := matrix(3, 3, [3, 1, -3, 0, 2, 6, -7, 1, 2]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"$\"\" \"7%F*!\"#F.7%!\"\"F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'ma trixG6#7%7%\"\"$\"\"\"!\"$7%\"\"!\"\"#\"\"'7%!\"(F+F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "( a) The square of the product of both matrices, " }{TEXT 505 1 "(" } {TEXT -1 1 "[" }{TEXT 503 1 "A" }{TEXT -1 3 "] [" }{TEXT 504 1 "B" } {TEXT -1 1 "]" }{TEXT 506 3 ")^2" }{TEXT -1 27 ", is the following ma trix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`(AB)^2` := evalm( (A &* B)^2) : (A*B)^`2` = matrix(`(AB)^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)*&%\"AG\"\"\"%\"BGF'%\"2G-%'matrixG6#7%7%\"#T\"#6\"#E 7%\"$?\"\"$3\"!\"'7%!#!)!#Q\"#h" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "(b) The product of either ma trix raised to the same power, [" }{TEXT 507 1 "A" }{TEXT -1 1 "]" } {TEXT 509 3 "^2 " }{TEXT -1 1 "[" }{TEXT 508 1 "B" }{TEXT -1 1 "]" } {TEXT 510 2 "^2" }{TEXT -1 27 ", is the following matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`A^2 B^2` := evalm(A^2 &* B^2) : \+ A^2 * B^2 = matrix(`A^2 B^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*& )%\"AG\"\"#\"\"\")%\"BGF'F(-%'matrixG6#7%7%\"$\"H\"#Z!$E\"7%!$K$\"#_\" $]#7%\"$!Q!#O!$`#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The resultant matrices of (a) and (b) are different." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "(2) Let " }{XPPEDIT 18 0 "n=2" "6#/%\"nG\"\"#" }{TEXT -1 7 " and " }{TEXT 512 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 514 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 515 1 ")" } {TEXT -1 22 " commuting matrices [" }{TEXT 511 1 "A" }{TEXT -1 7 "] a nd [" }{TEXT 513 1 "B" }{TEXT -1 45 "] be the same that are used in Un it (4), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "A := matri x(2, 2, [6, 8, 4, 6]) : B := matrix(2, 2, [15, 20, 10, 15]) : A = \+ matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"A G-%'matrixG6#7$7$\"\"'\"\")7$\"\"%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"BG-%'matrixG6#7$7$\"#:\"#?7$\"#5F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "(a) The square of the product of both matrices, " }{TEXT 518 1 "(" }{TEXT -1 1 "[" } {TEXT 516 1 "A" }{TEXT -1 3 "] [" }{TEXT 517 1 "B" }{TEXT -1 1 "]" } {TEXT 519 3 ")^2" }{TEXT -1 27 ", is the following matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`(AB)^2` := evalm((A &* B)^2 ) : (A*B)^`2` = matrix(`(AB)^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/)*&%\"AG\"\"\"%\"BGF'%\"2G-%'matrixG6#7$7$\"&+x&\"&+;)7$\"&+3%F/" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "(b) The product of either matrix raised to the same power , [" }{TEXT 520 1 "A" }{TEXT -1 1 "]" }{TEXT 522 3 "^2 " }{TEXT -1 1 "[" }{TEXT 521 1 "B" }{TEXT -1 1 "]" }{TEXT 523 2 "^2" }{TEXT -1 27 ", is the following matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`A^2 B^2` := evalm(A^2 &* B^2) : A^2 * B^2 = matrix(`A^2 B^2`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"AG\"\"#\"\"\")%\"BGF'F(-%'m atrixG6#7$7$\"&+x&\"&+;)7$\"&+3%F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The resultant matrices \+ of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "(3) Let " }{XPPEDIT 18 0 "n=2" "6# /%\"nG\"\"#" }{TEXT -1 7 " and " }{TEXT 525 1 "(" }{XPPEDIT 18 0 "3 " "6#\"\"$" }{TEXT 527 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 528 1 ")" }{TEXT -1 21 " diagonal matrices [" }{TEXT 524 1 "A" } {TEXT -1 7 "] and [" }{TEXT 526 1 "B" }{TEXT -1 13 "] be given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "A := diag(3, -5, 7) : B := diag(-2, 4, 6) : A =matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"$\"\"!F+7%F+!\"&F+7%F+F+\" \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%!\"#\"\"! F+7%F+\"\"%F+7%F+F+\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "(a) The square of the product of b oth matrices, " }{TEXT 531 1 "(" }{TEXT -1 1 "[" }{TEXT 529 1 "A" } {TEXT -1 3 "] [" }{TEXT 530 1 "B" }{TEXT -1 1 "]" }{TEXT 532 3 ")^2" } {TEXT -1 27 ", is the following matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`(AB)^2` := evalm((A &* B)^2) : (A*B)^`2` = matrix( `(AB)^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)*&%\"AG\"\"\"%\"BGF'% \"2G-%'matrixG6#7%7%\"#O\"\"!F07%F0\"$+%F07%F0F0\"%k<" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "( b) The product of either matrix raised to the same power, [" }{TEXT 533 1 "A" }{TEXT -1 1 "]" }{TEXT 535 3 "^2 " }{TEXT -1 1 "[" }{TEXT 534 1 "B" }{TEXT -1 1 "]" }{TEXT 536 2 "^2" }{TEXT -1 27 ", is the fo llowing matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`A^2 B^2 ` := evalm(A^2 &* B^2) : A^2 * B^2 = matrix(`A^2 B^2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"AG\"\"#\"\"\")%\"BGF'F(-%'matrixG6#7% 7%\"#O\"\"!F17%F1\"$+%F17%F1F1\"%k<" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The resultant matrices \+ of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 286 "" 0 "" {TEXT 537 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 459 4 "N.B." }{TEXT -1 7 " The " }{TEXT 556 4 "zero" }{TEXT -1 24 " matrix raised to an y " }{TEXT 460 8 "positive" }{TEXT -1 40 " integer power is the matr ix unchanged." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "As an example, consider the " }{TEXT 462 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 463 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 464 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 461 1 "0" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`0` := matrix(2, 2, [0, 0, 0, 0]) : `0` = matrix(`0 `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"0G-%'matrixG6#7$7$\"\"!F*F )" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The square of matrix [" }{TEXT 539 1 "0" }{TEXT -1 26 "] \+ is the following matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`0^2` := evalm(`0`^2) : `0`^2 = matrix(`0^2`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*$)%\"0G\"\"#\"\"\"-%'matrixG6#7$7$\"\"!F.F-" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 282 "" 0 "" {TEXT 465 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 466 4 "N.B." }{TEXT -1 46 " Literature sources \+ use a convention that a " }{TEXT 467 13 "square matrix" }{TEXT -1 23 " raised to the power " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 39 " is \+ equal to the appropriately sized " }{TEXT 468 11 "unit matrix" } {TEXT -1 7 ", i.e." }}}{EXCHG {PARA 283 "" 0 "" {TEXT -1 1 "[" } {TEXT 469 1 "A" }{TEXT -1 1 "]" }{TEXT 470 1 "^" }{XPPEDIT 18 0 "0" "6 #\"\"!" }{TEXT 472 3 " = " }{TEXT -1 1 "[" }{TEXT 471 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 3 "In " }{TEXT 473 5 "Maple" }{TEXT -1 44 ", the result of \+ this exponentiation is the " }{TEXT 578 6 "scalar" }{TEXT -1 9 " val ue " }{XPPMATH 20 "6#%$oneG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, co nsider a " }{TEXT 475 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 476 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 477 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 474 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(2, 2, [a[11], a[12], a[ 21], a[22]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Raising matrix [" }{TEXT 540 1 "A" }{TEXT -1 16 "] to the power " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 8 " yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "`A^0` := evalm(A^0) : A ^ `0` = `A^0` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%\"0G\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 284 "" 0 "" {TEXT 478 5 "* * * " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 3 "B. " }{TEXT 266 34 "Negative-integer power of a matrix" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "The negative-integer power of a square matrix of order " } {TEXT 435 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 436 3 " \327 " } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 437 1 ")" }{TEXT -1 6 " or " } {TEXT 438 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 439 3 " \327 " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 440 1 ")" }{TEXT -1 119 " is a mat rix of the same order. The resultant is a product matrix that is obtai ned by multiple multiplication of the " }{TEXT 263 7 "inverse" } {TEXT -1 73 " of a given matrix. For example, the negative second pow er of a matrix [" }{TEXT 264 1 "A" }{TEXT -1 28 "] is defined by the r elation" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 1 "[" }{TEXT 271 1 "A" } {TEXT -1 1 "]" }{TEXT 272 2 "^(" }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT 441 8 ") = (Inv" }{TEXT -1 1 "[" }{TEXT 265 1 "A" }{TEXT -1 1 " ]" }{TEXT 442 2 ")^" }{XPPEDIT 18 0 "2" "6#\"\"#" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplaril y, consider a " }{TEXT 443 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 444 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 445 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 309 1 "A" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(2, 2, [a[11], a[ 12], a[21], a[22]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The negative second power of [" }{TEXT 310 1 "A" }{TEXT -1 68 "] may be obtained using either of the following alternative met hods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 367 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 368 5 "ev alm" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`A^(-2)` := evalm(A^(-2)) : A ^ ` -2` = matrix(`A^(-2)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-2G-%'matrixG6#7$7$,&*&*$)&% \"aG6#\"#A\"\"#\"\"\"F5*$),&*&&F16#\"#6F5F0F5!\"\"*&&F16#\"#7F5&F16#\" #@F5F5F4F5F=F5*&*&F?F5FBF5F5*$F7F5F=F5,&*&*&F0F5F?F5F5*$F7F5F=F=*&*&F? F5F:F5F5*$F7F5F=F=7$,&*&*&FBF5F0F5F5*$F7F5F=F=*&*&F:F5FBF5F5*$F7F5F=F= ,&FEF5*&*$)F:F4F5F5*$F7F5F=F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "or, in a more compact form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`A^(-2)` := map(x->normal (x, expanded), `A^(-2)`) : A ^ ` -2` = matrix(`A^(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-2G-%'matrixG6#7$7$*&,&*$)&%\"aG6# \"#A\"\"#\"\"\"F5*&&F16#\"#7F5&F16#\"#@F5F5F5,(*&)&F16#\"#6F4F5F/F5F5* ,F4F5F@F5F0F5F7F5F:F5!\"\"*&)F7F4F5)F:F4F5F5FD*&,&*&F0F5F7F5FD*&F7F5F@ F5FDF5F=FD7$*&,&*&F:F5F0F5FD*&F@F5F:F5FDF5F=FD*&,&F6F5*$F?F5F5F5F=FD" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 371 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 369 5 "evalm " }{TEXT -1 5 " and " }{TEXT 370 7 "inverse" }{TEXT -1 11 " functions: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "`A^(-2)` := evalm((inve rse(A))^2) : A ^ ` -2` = matrix(`A^(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-2G-%'matrixG6#7$7$,&*&*$)&%\"aG6#\"#A\"\"#\" \"\"F5*$),&*&&F16#\"#6F5F0F5!\"\"*&&F16#\"#7F5&F16#\"#@F5F5F4F5F=F5*&* &F?F5FBF5F5*$F7F5F=F5,&*&*&F0F5F?F5F5*$F7F5F=F=*&*&F?F5F:F5F5*$F7F5F=F =7$,&*&*&FBF5F0F5F5*$F7F5F=F=*&*&F:F5FBF5F5*$F7F5F=F=,&FEF5*&*$)F:F4F5 F5*$F7F5F=F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "or, in a compact form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`A^(-2)` := map(x->normal(x, expanded), `A^(-2)` ) : A ^ ` -2` = matrix(`A^(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/)%\"AG%$~-2G-%'matrixG6#7$7$*&,&*$)&%\"aG6#\"#A\"\"#\"\"\"F5*&&F16# \"#7F5&F16#\"#@F5F5F5,(*&)&F16#\"#6F4F5F/F5F5*,F4F5F@F5F0F5F7F5F:F5!\" \"*&)F7F4F5)F:F4F5F5FD*&,&*&F0F5F7F5FD*&F7F5F@F5FDF5F=FD7$*&,&*&F:F5F0 F5FD*&F@F5F:F5FDF5F=FD*&,&F6F5*$F?F5F5F5F=FD" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "As a numerical \+ example, compute the negative second power of a " }{TEXT 446 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 447 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 448 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 311 1 "A" } {TEXT -1 10 "] given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " A := matrix(2, 2, [-2, 4, 6, 1]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$!\"#\"\"%7$\"\"'\"\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The negative second power of [" }{TEXT 312 1 "A" }{TEXT -1 20 " ] is the following " }{TEXT 449 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 450 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 451 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`A ^(-2)` := evalm(A^(-2)) : A ^ ` -2` = matrix(`A^(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-2G-%'matrixG6#7$7$#\"#D\"$w'#\"\" \"\"$p\"7$#\"\"$\"$Q$#\"\"(F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "`A^(-2)` := evalm((inverse(A))^2) : A ^ ` -2` = m atrix(`A^(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-2G-%'m atrixG6#7$7$#\"#D\"$w'#\"\"\"\"$p\"7$#\"\"$\"$Q$#\"\"(F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "F loating-point evaluation of the negative second power of the matrix [ " }{TEXT 313 1 "A" }{TEXT -1 25 "] gives the approximation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`A^(-2)` := evalf(evalm(`A^(-2)`)) \+ : A ^ ` -2` = matrix(`A^(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /)%\"AG%$~-2G-%'matrixG6#7$7$$\"+_[A)p$!#6$\"+j(fr\"f!#77$$\"+X'Rd())F 1$\"+M=,UTF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 268 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 557 4 "N.B." }{TEXT -1 7 " The " } {TEXT 558 4 "unit" }{TEXT -1 84 " matrix raised to any (positive or n egative) integer power is the matrix unchanged." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Exemplarily, compute the square and power " }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" }{TEXT -1 10 " of the " }{TEXT 560 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"# " }{TEXT 561 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 562 1 ")" }{TEXT -1 15 " unit matrix [" }{TEXT 559 1 "U" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U := diag(1, 1) : U = matr ix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7$7$\"\" \"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "`U^2` := evalm(U^2) : `U^(-3)` := evalm(U^(-3)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "U^2 = m atrix(`U^2`) ; U ^ ` -3` = matrix(`U^(-3)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"UG\"\"#\"\"\"-%'matrixG6#7$7$F(\"\"!7$F.F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"UG%$~-3G-%'matrixG6#7$7$\"\"\"\" \"!7$F-F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 287 "" 0 "" {TEXT 563 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (16) for \"" }{TEXT 458 36 "The complex matrix and its conjugate" }{TEXT -1 2 "\". " }}}{EXCHG {PARA 263 "" 0 "" {TEXT 457 67 "-------------------------- -----------------------------------------" }}}}{MARK "5 0" 0 } {VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }