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" " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 270 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 271 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 272 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT 444 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 274 "" 0 "" {TEXT 446 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT 447 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(13)" }{TEXT 304 1 " " }{TEXT 303 23 "The adjoint of a \+ matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 469 10 "OBJECTIVES" }{TEXT 470 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 471 1 "\225" } {TEXT -1 17 " To define the " }{TEXT 472 7 "adjoint" }{TEXT -1 6 " \+ or " }{TEXT 477 8 "adjugate" }{TEXT -1 21 " of a square matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 473 1 "\225" }{TEXT -1 70 " To provide a lternative methods of computing the adjoint of a matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 475 1 "\225" }{TEXT -1 33 " To introduce the co ncept of a " }{TEXT 474 15 "cofactor matrix" }{TEXT -1 91 " associat ed with a matrix and show how it may be used to obtain the adjoint of \+ the matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 476 1 "\225" }{TEXT -1 66 " To specify and illustrate some properties of the adjoint matrix. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 478 1 "\225" }{TEXT -1 17 " To defi ne the " }{TEXT 479 12 "self-adjoint" }{TEXT -1 40 " matrix and inve stigate its properties." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "restart : with(linalg, ad j, adjoint, coldim, det, diag, inverse, minor, multiply, rowdim, scala rmul, transpose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 261 7 "adjoint" }{TEXT -1 6 " or " }{TEXT 262 8 "adjugate" }{TEXT -1 22 " of a matrix is the \+ " }{TEXT 258 9 "transpose" }{TEXT -1 60 " of the matrix, obtained by \+ replacing each element by its " }{TEXT 259 8 "cofactor" }{TEXT -1 4 " ( " }{TEXT 260 12 "signed minor" }{TEXT -1 3 " )." }}{PARA 2 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The adjoint of a matrix is defined " }{TEXT 263 4 "only" }{TEXT -1 6 " for " }{TEXT 264 15 " square matrices" }{TEXT -1 61 ", irrespective of whether or not the m atrix is non-singular." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 389 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 265 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 390 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 266 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A := matrix(3, 3, [a[11], a[12], a[13], a[21], a[22], a[23], a[31], a [32], a[33]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"AG-%'matrixG6#7%7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"# A&F+6#\"#B7%&F+6#\"#J&F+6#\"#K&F+6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The matrix " } {TEXT 267 7 "adjoint" }{TEXT -1 13 " to matrix [" }{TEXT 268 1 "A" } {TEXT -1 8 "] is a " }{TEXT 391 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 289 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 392 1 ")" } {TEXT -1 82 " matrix, which can be obtained using either of the follo wing alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 305 8 "Method 1" }{TEXT -1 12 ". Using th e " }{TEXT 306 7 "adjoint" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`adj(A)` := adjoint(A) : Adj(A) = matri x(`adj(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AdjG6#%\"AG-%'mat rixG6#7%7%,&*&&%\"aG6#\"#A\"\"\"&F06#\"#LF3F3*&&F06#\"#BF3&F06#\"#KF3! \"\",&*&&F06#\"#7F3F4F3F>*&&F06#\"#8F3F;F3F3,&*&FAF3F8F3F3*&FEF3F/F3F> 7%,&*&&F06#\"#@F3F4F3F>*&F8F3&F06#\"#JF3F3,&*&&F06#\"#6F3F4F3F3*&FEF3F RF3F>,&*&FWF3F8F3F>*&FEF3FNF3F37%,&*&FNF3F;F3F3*&F/F3FRF3F>,&*&FWF3F;F 3F>*&FAF3FRF3F3,&*&FWF3F/F3F3*&FAF3FNF3F>" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 307 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 308 3 "adj" }{TEXT -1 10 " function:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`adj(A)` := adj(A) : Adj(A ) = matrix(`adj(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AdjG6#% \"AG-%'matrixG6#7%7%,&*&&%\"aG6#\"#A\"\"\"&F06#\"#LF3F3*&&F06#\"#BF3&F 06#\"#KF3!\"\",&*&&F06#\"#7F3F4F3F>*&&F06#\"#8F3F;F3F3,&*&FAF3F8F3F3*& FEF3F/F3F>7%,&*&&F06#\"#@F3F4F3F>*&F8F3&F06#\"#JF3F3,&*&&F06#\"#6F3F4F 3F3*&FEF3FRF3F>,&*&FWF3F8F3F>*&FEF3FNF3F37%,&*&FNF3F;F3F3*&F/F3FRF3F>, &*&FWF3F;F3F>*&FAF3FRF3F3,&*&FWF3F/F3F3*&FAF3FNF3F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exempla rily, consider a " }{TEXT 393 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 269 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 394 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 270 1 "B" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "B := matrix(3, 3, [1, 2, 1, 3, 1, 0, 2, 1, 2]) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"\"\"\"#F*7%\"\"$F*\"\"!7%F+F*F+" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The adjoint of matrix [" }{TEXT 271 1 "B" }{TEXT -1 20 "] is th e following " }{TEXT 395 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 302 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 396 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`adj(B)` := adjoint(B) : Adj(B) = matrix(`adj(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AdjG6#%\"BG-%'matrixG6#7%7%\"\"#!\"$!\"\"7%!\"'\"\" !\"\"$7%\"\"\"F3!\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 259 "" 0 "" {TEXT 288 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 4 "N.B." }{TEXT -1 66 " A matrix obtained by replacing each element of a square matri x [" }{TEXT 311 1 "A" }{TEXT -1 10 "] by its " }{TEXT 310 8 "cofactor " }{TEXT -1 31 " is called by some sources a " }{TEXT 313 15 "cofact or matrix" }{TEXT -1 19 " associated with [" }{TEXT 312 1 "A" }{TEXT -1 85 "]. The cofactor matrix may be used to obtain both the matrix ad joint and inverse to [" }{TEXT 317 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " }{TEXT 468 7 "inverse" }{TEXT -1 36 " of a matrix, refer \+ to Unit (14). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 397 1 " (" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 315 3 " \327 " }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 398 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 316 1 " A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(3, 3, [2, 3, 4, -5, 5, 6, 7, 8, 9]) : A = matrix(A) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"$ \"\"%7%!\"&\"\"&\"\"'7%\"\"(\"\")\"\"*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "and obtain the ma trix adjoint to [" }{TEXT 319 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 318 6 "Step 1 " }{TEXT -1 47 ". Declare the cofactor matrix associated with [" } {TEXT 320 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "cofactor(A) := matrix(3, 3) :" }}{PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 322 6 "Step 2" }{TEXT -1 40 ". Compu te cofactors of each element of [" }{TEXT 321 1 "A" }{TEXT -1 19 "] us ing the double " }{TEXT 323 3 "for" }{TEXT -1 16 "-loop construct:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "for i to rowdim(A) do for j to coldim(A) do cofactor(A)[i,j] := (-1)^(i+j)*det(minor(A, i, j)) : print(Cofactor(a[i,j]) = cofactor(A)[i,j]) : od : od :" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\"aG6$\"\"\"F*!\"$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\"aG6$\"\"\"\"\"#\"# ()" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\"aG6$\"\"\"\" \"$!#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\"aG6$\"\"# \"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\"aG6$ \"\"#F*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\"aG6$ \"\"#\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\" aG6$\"\"$\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6# &%\"aG6$\"\"$\"\"#!#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG 6#&%\"aG6$\"\"$F*\"#D" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 324 6 "Step 3" }{TEXT -1 36 ". Input/disp lay the cofactor matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "cofactor(A) := matrix(cofactor(A)) : Cofactor(A) = matrix(cofactor( A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#%\"AG-%'matrix G6#7%7%!\"$\"#()!#v7%\"\"&!#5F17%!\"#!#K\"#D" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 325 6 "Step 4" }{TEXT -1 47 ". Compute the transpose of the cofactor matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`transp(cofactor(A))` := transpose( cofactor(A)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Transp(Co factor(A)) = matrix(`transp(cofactor(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#-%)CofactorG6#%\"AG-%'matrixG6#7%7%!\"$\" \"&!\"#7%\"#()!#5!#K7%!#vF1\"#D" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 327 6 "Step 5" }{TEXT -1 57 ". Veri fy this result by computing the matrix adjoint to [" }{TEXT 326 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`adj(A)` := adjoint(A) : Adj(A) = matrix(`adj(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AdjG6#%\"AG-%'matrixG6#7%7%!\"$\"\"&!\"#7%\"#()!#5! #K7%!#vF.\"#D" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Both matrices of Steps 4 and 5 are equal. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 328 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 330 4 "N.B." }{TEXT -1 34 " The product of a sq uare matrix [" }{TEXT 331 1 "A" }{TEXT -1 56 "] and transpose of the c ofactor matrix associated with [" }{TEXT 332 1 "A" }{TEXT -1 44 "] equ als the product of the determinant of [" }{TEXT 333 1 "A" }{TEXT -1 43 "] and the unit matrix, which results in a " }{TEXT 273 6 "scalar " }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 1 "[" } {TEXT 334 1 "A" }{TEXT -1 2 "] " }{TEXT 338 15 "Transp(Cofactor" } {TEXT -1 1 "[" }{TEXT 335 1 "A" }{TEXT -1 1 "]" }{TEXT 399 8 ") = (Det " }{TEXT -1 1 "[" }{TEXT 336 1 "A" }{TEXT -1 1 "]" }{TEXT 400 1 ")" } {TEXT -1 2 " [" }{TEXT 337 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Exemplarily, consider the same " }{TEXT 401 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 339 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 402 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 340 1 "A" }{TEXT -1 12 "] as before." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(a) The product [" }{TEXT 342 1 "A" }{TEXT -1 2 "] " } {TEXT 344 15 "Transp(Cofactor" }{TEXT -1 1 "[" }{TEXT 343 1 "A" } {TEXT -1 1 "]" }{TEXT 403 1 ")" }{TEXT -1 20 " is the following " } {TEXT 404 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 341 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 405 1 ")" }{TEXT -1 2 " " }{TEXT 345 6 "scalar" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`A transp(cofactor(A))` := evalm(A &* transpose(cofac tor(A))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A*Transp(Cofa ctor(A)) = matrix(`A transp(cofactor(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"-%'TranspG6#-%)CofactorG6#F%F&-%'matrixG6 #7%7%!#X\"\"!F37%F3F2F37%F3F3F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(b) The unit matrix [" } {TEXT 351 1 "U" }{TEXT -1 38 "] appropriately sized for this case is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U := diag(1, 1, 1) : U \+ = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7%7 %\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(c) The product (" }{TEXT 347 3 "Det" }{TEXT -1 1 "[" }{TEXT 346 1 "A" }{TEXT -1 1 "]" }{TEXT 406 1 ")" }{TEXT -1 2 " [" }{TEXT 348 1 "U" }{TEXT -1 21 "] is the fo llowing " }{TEXT 407 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 349 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 408 1 ")" }{TEXT -1 2 " \+ " }{TEXT 350 6 "scalar" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "`det(A) U` := evalm(det(A) * U) : Det(A)*U = matrix(`det(A) U`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$DetG6#% \"AG\"\"\"%\"UGF)-%'matrixG6#7%7%!#X\"\"!F17%F1F0F17%F1F1F0" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Both product matrices of (a) and (c) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 329 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 4 "N.B." }{TEXT -1 42 " The product of a square matrix and its " }{TEXT 274 7 "adjoint" }{TEXT -1 13 " obeys the " } {TEXT 275 15 "commutative law" }{TEXT -1 30 ". The product matrix is \+ the " }{TEXT 276 6 "unit " }{TEXT -1 79 "matrix multiplied by the de terminant of the square matrix, which results in a " }{TEXT 460 6 "sc alar" }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 411 4 " (Adj" }{TEXT -1 1 "[" }{TEXT 277 1 "A" }{TEXT -1 1 "]" }{TEXT 409 1 ") " }{TEXT -1 2 " [" }{TEXT 278 1 "A" }{TEXT -1 1 "]" }{TEXT 410 3 " = \+ " }{TEXT -1 1 "[" }{TEXT 279 1 "A" }{TEXT -1 2 "] " }{TEXT 412 4 "(Adj " }{TEXT -1 1 "[" }{TEXT 280 1 "A" }{TEXT -1 1 "]" }{TEXT 413 8 ") = ( Det" }{TEXT -1 1 "[" }{TEXT 281 1 "A" }{TEXT -1 1 "]" }{TEXT 414 1 ") " }{TEXT -1 2 " [" }{TEXT 353 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplari ly, consider a " }{TEXT 415 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 282 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 416 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 283 1 "A" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A := matrix(3, 3, [1, 2, 3, 4, 0, 1, 2, 3, 5]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"!F*7%F+F,\"\"& " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The adjoint of the matrix, " }{TEXT 301 3 "Adj" }{TEXT -1 1 "[" }{TEXT 290 1 "A" }{TEXT -1 22 "], is the following " } {TEXT 417 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 292 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 418 1 ")" }{TEXT -1 9 " matrix:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`adj(A)` := adjoint(A) : \+ Adj(A) = matrix(`adj(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Ad jG6#%\"AG-%'matrixG6#7%7%!\"$!\"\"\"\"#7%!#=F.\"#67%\"#7\"\"\"!\")" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(a) The product " }{TEXT 419 4 "(Adj" }{TEXT -1 1 "[" }{TEXT 284 1 "A" }{TEXT -1 1 "]" }{TEXT 420 1 ")" }{TEXT -1 2 " [" }{TEXT 285 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 421 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 293 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 422 1 ")" }{TEXT -1 2 " " }{TEXT 294 6 "scalar" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`a dj(A) A` := multiply(`adj(A)`, A) : Adj(A) * A = matrix(`adj(A) A`) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$AdjG6#%\"AG\"\"\"F(F)-%'mat rixG6#7%7%!\"$\"\"!F07%F0F/F07%F0F0F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product \+ [" }{TEXT 286 1 "A" }{TEXT -1 2 "] " }{TEXT 423 4 "(Adj" }{TEXT -1 1 " [" }{TEXT 287 1 "A" }{TEXT -1 1 "]" }{TEXT 424 1 ")" }{TEXT -1 20 " i s the following " }{TEXT 425 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 295 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 426 1 ")" } {TEXT -1 2 " " }{TEXT 296 6 "scalar" }{TEXT -1 9 " matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`A adj(A)` := multiply(A, `a dj(A)`) : A * `Adj(A)` = matrix(`A adj(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%'Adj(A)GF&-%'matrixG6#7%7%!\"$\"\"!F.7%F .F-F.7%F.F.F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "(c) With the appropriately sized unit mat rix, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "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 24 "and the determinant of [" }{TEXT 297 1 "A " }{TEXT -1 11 "], which is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "the product " }{TEXT 427 4 "(Det" }{TEXT -1 1 "[" }{TEXT 298 1 "A" }{TEXT -1 1 "]" }{TEXT 428 1 ")" }{TEXT -1 2 " [" }{TEXT 352 1 "U" }{TEXT -1 21 "] is the fo llowing " }{TEXT 429 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 299 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 430 1 ")" }{TEXT -1 2 " \+ " }{TEXT 300 6 "scalar" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "`det(A) U` := scalarmul(U, det(A)) : Det(A)* U = matrix(`det(A) U`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$DetG 6#%\"AG\"\"\"%\"UGF)-%'matrixG6#7%7%!\"$\"\"!F17%F1F0F17%F1F1F0" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "All three product matrices are equal." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 291 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 371 4 "N.B." }{TEXT -1 111 " The adjoint of the product of two matric es is the product of the two matrix adjoints but in the reverse order " }}}{EXCHG {PARA 265 "" 0 "" {TEXT 363 4 "Adj(" }{TEXT -1 1 "[" } {TEXT 367 1 "A" }{TEXT -1 3 "] [" }{TEXT 368 1 "B" }{TEXT -1 1 "]" } {TEXT 364 1 ")" }{TEXT -1 1 " " }{TEXT 362 1 "=" }{TEXT -1 1 " " } {TEXT 365 3 "Adj" }{TEXT -1 1 "[" }{TEXT 369 1 "B" }{TEXT -1 1 "]" } {TEXT 366 4 " Adj" }{TEXT -1 1 "[" }{TEXT 370 1 "A" }{TEXT -1 1 "]" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 27 "Exemplarily, consider two " }{TEXT 431 1 "(" }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 360 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 361 1 ")" }{TEXT 432 12 " matrices [" }{TEXT 356 1 "A" }{TEXT 358 7 "] and [" }{TEXT 357 1 "B" }{TEXT 359 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "A := matrix(3, 3, [1, -2, 3, 4, 7, 1, 2, 3, -4]) : B := matrix(3, 3, [2, 5, -1, 3, 4, 0, 6, -1, 2]) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = m atrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\" \"\"!\"#\"\"$7%\"\"%\"\"(F*7%\"\"#F,!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"#\"\"&!\"\"7%\"\"$\"\"%\"\"!7%\"\"'F,F *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "(a) The adjoint of the product of the two matrices, " } {TEXT 372 4 "Adj(" }{TEXT -1 1 "[" }{TEXT 374 1 "A" }{TEXT -1 3 "] [" }{TEXT 375 1 "B" }{TEXT -1 1 "]" }{TEXT 373 1 ")" }{TEXT -1 21 ", is \+ the following " }{TEXT 433 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 434 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 435 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`adj(AB) ` := adjoint(multiply(A, B)) : Adj(A * B) = matrix(`adj(AB)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AdjG6#*&%\"AG\"\"\"%\"BGF)-%'matri xG6#7%7%!$=%\"#q!$B#7%\"$s$!#&)\"$.#7%\"%F9!$)H\"$o)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "(b) The product of the two matrix adjoints multiplied in the reverse order, \+ " }{TEXT 376 3 "Adj" }{TEXT -1 1 "[" }{TEXT 378 1 "B" }{TEXT -1 1 "]" }{TEXT 377 4 " Adj" }{TEXT -1 1 "[" }{TEXT 379 1 "A" }{TEXT -1 22 "], \+ is the following " }{TEXT 436 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 437 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 438 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "`a dj(B) adj(A)` := multiply(adjoint(B), adjoint(A)) : Adj(B) * Adj(A) \+ = matrix(`adj(B) adj(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$A djG6#%\"BG\"\"\"-F&6#%\"AGF)-%'matrixG6#7%7%!$=%\"#q!$B#7%\"$s$!#&)\"$ .#7%\"%F9!$)H\"$o)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 380 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 309 4 "N.B." }{TEXT -1 54 " A matrix that equals its own adjoint is said to be " }{TEXT 354 12 "se lf-adjoint" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " } {TEXT 439 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 440 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 441 1 ")" }{TEXT -1 8 " matrix" } {TEXT 452 2 " [" }{TEXT 450 1 "A" }{TEXT 451 2 "] " }{TEXT -1 8 "given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A := matrix(3, 3, [- 4, -3, -3, 1, 0, 1, 4, 4, 3]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%!\"%!\"$F+7%\"\"\"\"\"!F-7%\"\"%F 0\"\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The adjoint of matrix" }{TEXT 455 2 " [" }{TEXT 453 1 "A" }{TEXT 454 2 "] " }{TEXT -1 18 "is the following " }{TEXT 456 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 457 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 458 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`adj(A)` := adjoint(A) : Adj(A) = matrix(`adj(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AdjG6#%\"AG -%'matrixG6#7%7%!\"%!\"$F.7%\"\"\"\"\"!F07%\"\"%F3\"\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "N otice that the " }{TEXT 461 11 "determinant" }{TEXT -1 31 " of a sel f-adjoint matrix is " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 7 ", viz. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : \+ Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG \"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 276 " " 0 "" {TEXT 449 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 381 4 "N.B." }{TEXT -1 13 " Raising a \+ " }{TEXT 382 12 "self-adjoint" }{TEXT -1 40 " matrix to any (positive or negative) " }{TEXT 383 3 "odd" }{TEXT -1 84 " power does not cha nge the matrix, while raising it to any (positive or negative) " } {TEXT 384 4 "even" }{TEXT -1 19 " power returns a " }{TEXT 388 11 "u nit matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Consider the same self-adjoint ma trix [" }{TEXT 385 1 "A" }{TEXT -1 12 "] as before." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(a) Rai sing [" }{TEXT 386 1 "A" }{TEXT -1 17 "] to the powers " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "-5" "6#,$\"\" &!\"\"" }{TEXT -1 25 " yields the same matrix:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 102 "`A^3` := evalm(A^3) : `A^(-5)` := evalm(A^( -5)) : A^3=matrix(`A^3`) ; A^` -5`=matrix(`A^(-5)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"$\"\"\"-%'matrixG6#7%7%!\"%!\"$F/7 %F(\"\"!F(7%\"\"%F3F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-5G -%'matrixG6#7%7%!\"%!\"$F-7%\"\"\"\"\"!F/7%\"\"%F2\"\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "I t follows immediately from the above that " }{TEXT 462 9 "inversion" }{TEXT -1 59 " of a self-adjoint matrix does not change the matrix, v iz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv(A)` := inverse( A) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$InvG6#%\"AG-%'matrixG6#7%7%!\"%!\"$F.7%\"\"\"\"\"!F07%\"\"%F3\"\"$ " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(b) Raising [" }{TEXT 387 1 "A" }{TEXT -1 17 "] to the po wers " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "-6" "6#,$\"\"'!\"\"" }{TEXT -1 39 " yields the corresponding un it matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "`A^4` := eva lm(A^4) : `A^(-6)` := evalm(A^(-6)) : A^4=matrix(`A^4`) ; A^` -6 `=matrix(`A^(-6)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"% \"\"\"-%'matrixG6#7%7%F(\"\"!F.7%F.F(F.7%F.F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-6G-%'matrixG6#7%7%\"\"\"\"\"!F-7%F-F,F-7%F-F -F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "[ Refer to Unit (14) for the matrix " }{TEXT 463 7 "in verse" }{TEXT -1 24 " and to Unit (15) for " }{TEXT 464 22 "integer \+ exponentiation" }{TEXT -1 16 " of matrices. ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Since the de terminant of both a " }{TEXT 465 12 "self-adjoint" }{TEXT -1 7 " and " }{TEXT 466 4 "unit" }{TEXT -1 13 " matrix is " }{XPPMATH 20 "6#% &unityG" }{TEXT -1 85 ", it follows from the above that the determina nt of a self-adjoint matrix raised to " }{TEXT 467 3 "any" }{TEXT -1 6 " non-" }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 20 " integer power is \+ " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 15 ". Exemplarily," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`det(A^3)` := det(`A^3`) : `det(A ^(-6))` := det(`A^(-6)`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Det(A^3) = `det(A^3)` ; Det(A^` -6`) = `det(A^(-6))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#*$)%\"AG\"\"$\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#)%\"AG%$~-6G\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 277 "" 0 "" {TEXT 459 5 "* * * " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (14) for \"" }{TEXT 448 23 "The inverse o f a matrix" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 261 "" 0 "" {TEXT 442 67 "------------------------------------------------------------------ -" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }