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{CSTYLE " " -1 -1 "" 0 1 4 0 0 0 1 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 280 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 283 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 284 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 285 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 286 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 287 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 288 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 273 "" 0 "" {TEXT 451 39 "MATRICES AND MATRIX OPE RATIONS: Unit 10" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 274 "" 0 "" {TEXT 453 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 452 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 280 "" 0 "" {TEXT 454 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT 455 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(10)" }{TEXT 345 1 " " }{TEXT 344 25 "The transpose of \+ a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 602 10 "OBJECTIVES" }{TEXT 603 1 ":" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 604 1 "\225" } {TEXT -1 17 " To define the " }{TEXT 605 9 "transpose" }{TEXT -1 62 " of a matrix and provide examples of transposition operation." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 611 1 "\225" }{TEXT -1 78 " To show the \+ effect of transposition operation on specific types of matrices." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 606 1 "\225" }{TEXT -1 74 " To specify a nd illustrate general properties of transposition operation." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 610 1 "\225" }{TEXT -1 46 " To stress an d show that transposition of a " }{TEXT 607 6 "vector" }{TEXT -1 7 " \+ with " }{TEXT 608 5 "Maple" }{TEXT -1 19 " does not yield a " } {TEXT 609 13 "column matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 612 1 "\225" }{TEXT -1 28 " To introduce the function " } {TEXT 616 5 "equal" }{TEXT -1 35 " for testing if matrices are equal. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "restart : with(linalg, diag, equal, multiply, trans pose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 258 9 "transpose" }{TEXT -1 21 " of a \+ matrix ( or " }{TEXT 259 17 "transposed matrix" }{TEXT -1 122 " ) i s the matrix resulting from interchanging the rows and columns in the \+ given matrix. If the given matrix is of order " }{TEXT 346 1 "(" } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 347 3 " \327 " }{XPPEDIT 18 0 "n" " 6#%\"nG" }{TEXT 348 1 ")" }{TEXT -1 13 ", then its " }{TEXT 267 9 "t ranspose" }{TEXT -1 26 " is the matrix of order " }{TEXT 349 1 "(" } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 350 3 " \327 " }{XPPEDIT 18 0 "m" " 6#%\"mG" }{TEXT 351 1 ")" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " } {TEXT 352 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 353 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 354 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 268 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A := matrix(3, 2, [ a[11], a[12], a[21], a[22], a[ 31], a[32] ]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"AG-%'matrixG6#7%7$&%\"aG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A7$&F+6# \"#J&F+6#\"#K" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The transpose of the matrix [" }{TEXT 269 1 "A" }{TEXT -1 20 "] is the following " }{TEXT 355 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 356 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 357 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`transp(A)` := transpose(A) : Transp(A) = m atrix(`transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#% \"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F.6#\"#@&F.6#\"#J7%&F.6#\"#7&F.6#\"#A &F.6#\"#K" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-boo k\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Transp(ma trix(A)) = matrix(`transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %'TranspG6#-%'matrixG6#7%7$&%\"aG6#\"#6&F-6#\"#77$&F-6#\"#@&F-6#\"#A7$ &F-6#\"#J&F-6#\"#K-F(6#7$7%F,F4F;7%F0F7F>" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 281 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 4 "N.B." }{TEXT -1 22 " The transpose of a " }{TEXT 264 15 "diag onal matrix" }{TEXT -1 35 " is the same as the matrix itself." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Exemplarily, the transpose of a " }{TEXT 358 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 359 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 360 1 ")" }{TEXT -1 19 " diagonal matrix [" }{TEXT 270 1 "B" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " B := diag(3, 0, 2, 5) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7&7&\"\"$\"\"!F+F+7&F+F+F+F+7&F+F+\" \"#F+7&F+F+F+\"\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "is the following " }{TEXT 361 1 "(" } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 362 3 " \327 " }{XPPEDIT 18 0 "4" " 6#\"\"%" }{TEXT 363 1 ")" }{TEXT -1 18 " diagonal matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`transp(B)` := transpose(B) : Tra nsp(B) = matrix(`transp(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%' TranspG6#%\"BG-%'matrixG6#7&7&\"\"$\"\"!F.F.7&F.F.F.F.7&F.F.\"\"#F.7&F .F.F.\"\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-bo ok\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Transp(m atrix(B)) = matrix(`transp(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%'TranspG6#-%'matrixG6#7&7&\"\"$\"\"!F-F-7&F-F-F-F-7&F-F-\"\"#F-7&F-F -F-\"\"&F'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 282 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 330 4 "N.B." }{TEXT -1 22 " Transpo sition of a " }{TEXT 332 13 "square matrix" }{TEXT -1 7 " does " } {TEXT 331 3 "not" }{TEXT -1 13 " change the " }{TEXT 333 11 "determin ant" }{TEXT -1 16 " of the matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " }{TEXT 601 11 "determinant" }{TEXT -1 36 " of a matrix, refer to Unit (11). \+ ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 272 "" 0 " " {TEXT 329 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 260 4 "N.B." }{TEXT -1 21 " Transpositio n does " }{TEXT 265 3 "not" }{TEXT -1 14 " change the s" }{TEXT 266 8 "ymmetric" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Exemplarily, the transp ose of a " }{TEXT 364 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 365 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 366 1 ")" }{TEXT -1 20 " symmetric matrix [" }{TEXT 271 1 "C" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "C := matrix(3, 3, [1, a, a^2 , a, a^2, 1, a^2, 1, a]) : C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7%7%\"\"\"%\"aG*$)F+\"\"#F*7%F+F,F*7% F,F*F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "is the following " }{TEXT 367 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 368 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 369 1 ")" }{TEXT -1 19 " symmetric matrix:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 65 "`transp(C)` := transpose(C) : Transp(C) = \+ matrix(`transp(C)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6# %\"CG-%'matrixG6#7%7%\"\"\"%\"aG*$)F.\"\"#F-7%F.F/F-7%F/F-F." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-book\" form, viz ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Transp(matrix(C)) = ma trix(`transp(C)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#-% 'matrixG6#7%7%\"\"\"%\"aG*$)F-\"\"#F,7%F-F.F,7%F.F,F-F'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 283 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 4 "N.B." }{TEXT -1 35 " Transposition changes sign of \+ a " }{TEXT 261 14 "skew-symmetric" }{TEXT -1 47 " matrix but the mat rix is otherwise unchanged." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Exemplarily, the transpose of a \+ " }{TEXT 370 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 371 3 " \327 \+ " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 372 1 ")" }{TEXT -1 25 " skew-s ymmetric matrix [" }{TEXT 272 1 "E" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "E := matrix(3, 3, [0, a, b, \+ -a, 0, c, -b, -c, 0]) : E = matrix(E) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7%7%\"\"!%\"aG%\"bG7%,$F+!\"\"F*%\"cG 7%,$F,F/,$F0F/F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "is the following " }{TEXT 373 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 374 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 375 1 ")" }{TEXT -1 24 " skew-symmetric matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`transp(E)` := transpose(E) \+ : Transp(E) = matrix(`transp(E)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#%\"EG-%'matrixG6#7%7%\"\"!,$%\"aG!\"\",$%\"bGF07%F/F-, $%\"cGF07%F2F5F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like -in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " Transp(matrix(E)) = matrix(`transp(E)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#-%'matrixG6#7%7%\"\"!%\"aG%\"bG7%,$F-!\"\" F,%\"cG7%,$F.F1,$F2F1F,-F(6#7%7%F,F0F47%F-F,F57%F.F2F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 284 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 4 "N.B." }{TEXT -1 80 " According to the definition of the transpose of a matrix, the transpose of a " }{TEXT 376 1 "(" } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 377 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 378 1 ")" }{TEXT -1 2 " " }{TEXT 274 10 "row matrix " }{TEXT -1 9 " is an " }{TEXT 379 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG " }{TEXT 380 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 381 1 ") " }{TEXT -1 2 " " }{TEXT 275 13 "column matrix" }{TEXT -1 13 ", in w hich " }{XPPEDIT 18 0 "m=n" "6#/%\"mG%\"nG" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Exemplarily, the transpose of a " }{TEXT 382 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 383 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$ " }{TEXT 384 1 ")" }{TEXT -1 14 " row matrix [" }{TEXT 276 2 "RM" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " RM := matrix(1, 3, [a, b, c]) : RM = matrix(RM) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%#RMG-%'matrixG6#7#7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "i s the following " }{TEXT 385 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 386 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 387 1 ")" } {TEXT -1 16 " column matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`transp(RM)` := transpose(RM) : Transp(RM) = matrix(`transp(RM )`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#%#RMG-%'matrixG6 #7%7#%\"aG7#%\"bG7#%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Transp(matrix(RM)) = matrix(`transp(RM)`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%'TranspG6#-%'matrixG6#7#7%%\"aG%\"bG%\"cG-F(6#7%7# F,7#F-7#F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 286 "" 0 "" {TEXT 515 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 509 4 "N.B." }{TEXT -1 81 " In some textbooks, the following 'shortcut notation' is used in relation to a " }{TEXT 540 6 "vector" }}}{EXCHG {PARA 285 "" 0 "" {XPPEDIT 18 0 "[ a,b,c]^T" "6#)7%%\"aG%\"bG%\"cG%\"TG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "to mean \"convert vecto r " }{XPPEDIT 18 0 "[a,b,c]" "6#7%%\"aG%\"bG%\"cG" }{TEXT -1 8 " to \+ a " }{TEXT 613 13 "column matrix" }{TEXT -1 20 " by transposition\". " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Let us analyse this operation in " }{TEXT 614 5 "Maple" } {TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "If the vector " }{XPPEDIT 18 0 "[a,b,c] " "6#7%%\"aG%\"bG%\"cG" }{TEXT -1 31 " is assigned the name RV, viz. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "RV := array([a, b, c]) \+ : RV = eval(RV) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RVG-%'vector G6#7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "then, transposition operation of [ " }{TEXT 615 2 "RV" }{TEXT -1 8 "] yields" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "`transp(RV)` := transpose(RV) : Transp(RV) = tran spose(eval(RV)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#%#RV G-%*transposeG6#-%'vectorG6#7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "In executing t his operation, " }{TEXT 511 5 "Maple" }{TEXT -1 16 " treats vector [" }{TEXT 510 2 "RV" }{TEXT -1 19 "] as if it were a " }{TEXT 513 13 "co lumn vector" }{TEXT -1 39 " and, consequently, transposition of [" } {TEXT 512 2 "RV" }{TEXT -1 13 "] yields a " }{TEXT 514 10 "row vecto r" }{TEXT -1 46 ". This is evidenced by the following results." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(a) The product [" }{TEXT 519 2 "RV" }{TEXT -1 2 "] " }{TEXT 517 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 516 2 "RV" }{TEXT -1 1 "]" } {TEXT 518 2 ")," }{TEXT -1 14 " which is a " }{TEXT 520 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 521 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 522 1 ")" }{TEXT -1 2 " " }{TEXT 523 9 "symmetric" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "`R V transp(RV)` := evalm(RV &* `transp(RV)`) : RV * Transp(RV) = eval( `RV transp(RV)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#RVG\"\"\"-% 'TranspG6#F%F&-%'matrixG6#7%7%*$)%\"aG\"\"#F&*&F1F&%\"bGF&*&F1F&%\"cGF &7%F3*$)F4F2F&*&F4F&F6F&7%F5F:*$)F6F2F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "This corresponds \+ to the result of multiplication of a " }{TEXT 533 13 "column matrix" }{TEXT -1 9 " and a " }{TEXT 534 10 "row matrix" }{TEXT -1 20 " \226 refer to Section " }{TEXT 538 1 "B" }{TEXT -1 13 " of Unit (5)." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(b) The product " }{TEXT 529 7 "(Transp" }{TEXT -1 1 "[" } {TEXT 528 2 "RV" }{TEXT -1 1 "]" }{TEXT 530 1 ")" }{TEXT -1 2 " [" } {TEXT 531 2 "RV" }{TEXT -1 1 "]" }{TEXT 532 1 "," }{TEXT -1 14 " whic h is a " }{TEXT 539 6 "scalar" }{TEXT -1 11 " (number):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "`(transp(RV)) RV` := evalm(`transp( RV)` &* RV) : `Transp(RV)` * RV = eval(`(transp(RV)) RV`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%+Transp(RV)G\"\"\"%#RVGF&,(*$)%\"a G\"\"#F&F&*$)%\"bGF,F&F&*$)%\"cGF,F&F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "This corresponds \+ to the result of multiplication of a " }{TEXT 535 10 "row matrix" } {TEXT -1 9 " and a " }{TEXT 536 13 "column matrix" }{TEXT -1 20 " \+ \226 refer to Section " }{TEXT 537 1 "A" }{TEXT -1 13 " of Unit (5)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Thus, transposition of a (row) " }{TEXT 524 6 "vector" } {TEXT -1 7 " does " }{TEXT 525 3 "not" }{TEXT -1 10 " yield a " } {TEXT 526 13 "column matrix" }{TEXT -1 5 " in " }{TEXT 527 5 "Maple" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 285 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 4 "N.B." }{TEXT -1 81 " A ccording to the definition of the transpose of a matrix, the transpose of an " }{TEXT 388 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 389 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 390 1 ")" }{TEXT -1 2 " \+ " }{TEXT 279 13 "column matrix" }{TEXT -1 8 " is a " }{TEXT 391 1 "( " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 392 3 " \327 " }{XPPEDIT 18 0 " n" "6#%\"nG" }{TEXT 393 1 ")" }{TEXT -1 2 " " }{TEXT 278 10 "row matr ix" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Exemplarily, the transpose of a " } {TEXT 394 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 395 3 " \327 " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 396 1 ")" }{TEXT -1 17 " column m atrix [" }{TEXT 280 2 "CM" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CM := matrix(3, 1, [d, e, f]) : CM = ma trix(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7%7#% \"dG7#%\"eG7#%\"fG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "is the following " }{TEXT 397 1 "(" } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 398 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 399 1 ")" }{TEXT -1 13 " row matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`transp(CM)` := transpose(CM) : T ransp(CM) = matrix(`transp(CM)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%'TranspG6#%#CMG-%'matrixG6#7#7%%\"dG%\"eG%\"fG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This opera tion may be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Transp(matrix(CM)) = matrix(`transp (CM)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#-%'matrixG6#7 %7#%\"dG7#%\"eG7#%\"fG-F(6#7#7%F,F.F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 286 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 485 4 "N.B." }{TEXT -1 44 " The transpose of the product of a matrix \+ [" }{TEXT 484 1 "A" }{TEXT -1 16 "] and a scalar " }{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 54 " equals the matrix transpose multiplied by \+ the scalar" }}}{EXCHG {PARA 284 "" 0 "" {TEXT 486 7 "Transp(" }{TEXT 488 1 "k" }{TEXT 489 1 " " }{TEXT -1 1 "[" }{TEXT 492 1 "A" }{TEXT -1 1 "]" }{TEXT 487 4 ") = " }{TEXT 490 1 "k" }{TEXT 491 7 " Transp" } {TEXT -1 1 "[" }{TEXT 493 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 495 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 496 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 497 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 494 1 "A" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A := matrix(2, 3, [ a[11], a [12], a[13], a[21], a[22], a[23] ]) : A = matrix(A) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6# \"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "(a) The transpose of the p roduct of the matrix [" }{TEXT 504 1 "A" }{TEXT -1 14 "] and scalar \+ " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 3 ", " }{TEXT 498 7 "Transp( " }{TEXT 500 1 "k" }{TEXT 501 1 " " }{TEXT -1 1 "[" }{TEXT 502 1 "A" } {TEXT -1 1 "]" }{TEXT 499 1 ")" }{TEXT -1 21 ", is the following " } {TEXT 503 7 "(3 \327 2)" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71 "`transp(kA)` := transpose(k*A) : Transp(k*A) = matrix(`transp(kA)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Trans pG6#*&%\"kG\"\"\"%\"AGF)-%'matrixG6#7%7$*&F(F)&%\"aG6#\"#6F)*&F(F)&F26 #\"#@F)7$*&F(F)&F26#\"#7F)*&F(F)&F26#\"#AF)7$*&F(F)&F26#\"#8F)*&F(F)&F 26#\"#BF)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 67 "(b) The product of the matrix transpose multiplied by the scalar, " }{TEXT 505 1 "k" }{TEXT 506 7 " Transp" }{TEXT -1 1 "[" }{TEXT 507 1 "A" }{TEXT -1 22 "], is the following " }{TEXT 508 7 "(3 \327 2)" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 84 "`k transp(A)` := evalm(k * transpose(A)) : k * Tr ansp(A) = matrix(`k transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *&%\"kG\"\"\"-%'TranspG6#%\"AGF&-%'matrixG6#7%7$*&F%F&&%\"aG6#\"#6F&*& F%F&&F26#\"#@F&7$*&F%F&&F26#\"#7F&*&F%F&&F26#\"#AF&7$*&F%F&&F26#\"#8F& *&F%F&&F26#\"#BF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 542 5 "equal" }{TEXT -1 64 " \+ function applied to the resultant matrices of (a) and (b), i.e." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "equal(`transp(kA)`, `k trans p(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "returns the " }{TEXT 617 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6 #%%trueG" }{TEXT -1 47 ", which verifies that both matrices are equal ." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 283 "" 0 " " {TEXT 483 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 459 4 "N.B." }{TEXT -1 37 " The product \+ of the transpose of a " }{TEXT 461 6 "square" }{TEXT -1 10 " matrix \+ [" }{TEXT 473 1 "A" }{TEXT -1 23 "] and the matrix is a " }{TEXT 460 9 "symmetric" }{TEXT -1 49 " matrix. The product of the same square m atrix [" }{TEXT 475 1 "A" }{TEXT -1 31 "] and its transpose is also a \+ " }{TEXT 474 9 "symmetric" }{TEXT -1 51 " matrix but both resultant \+ matrices are different." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 463 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 464 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 465 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 462 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := matrix(2, 2, [a, b, c, d]) : A = matrix(A) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$%\"aG%\"bG7 $%\"cG%\"dG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(a) The product " }{TEXT 467 7 "(Transp" } {TEXT -1 1 "[" }{TEXT 466 1 "A" }{TEXT -1 1 "]" }{TEXT 471 1 ")" } {TEXT -1 2 " [" }{TEXT 472 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 468 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 469 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 470 1 ")" }{TEXT -1 19 " symmetric matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "`(transp(A)) A` := evalm(transpose(A) &* A) : Transp(A) * A = matrix(`(transp(A)) A `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'TranspG6#%\"AG\"\"\"F(F) -%'matrixG6#7$7$,&*$)%\"aG\"\"#F)F)*$)%\"cGF3F)F),&*&F2F)%\"bGF)F)*&F6 F)%\"dGF)F)7$F7,&*$)F9F3F)F)*$)F;F3F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product \+ [" }{TEXT 482 1 "A" }{TEXT -1 2 "] " }{TEXT 477 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 476 1 "A" }{TEXT -1 1 "]" }{TEXT 481 1 ")" }{TEXT -1 20 " is the following " }{TEXT 478 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 479 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 480 1 ")" } {TEXT -1 19 " symmetric matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`A transp(A)` := evalm(A &* transpose(A)) : A * `Tr ansp(A)` = matrix(`A transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&%\"AG\"\"\"%*Transp(A)GF&-%'matrixG6#7$7$,&*$)%\"aG\"\"#F&F&*$)%\"b GF1F&F&,&*&F0F&%\"cGF&F&*&F4F&%\"dGF&F&7$F5,&*$)F7F1F&F&*$)F9F1F&F&" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "The matrices of (a) and (b) are symmetric but different. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 282 "" 0 "" {TEXT 458 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 4 "N.B." }{TEXT -1 53 " The sum of a square matrix and its transpose is a " }{TEXT 294 9 "symmetric" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 400 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 401 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 402 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 288 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 10 "The sum [" }{TEXT 289 1 "A" } {TEXT -1 1 "]" }{TEXT 403 9 " + Transp" }{TEXT -1 1 "[" }{TEXT 290 1 " A" }{TEXT -1 21 "] is the following " }{TEXT 404 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 405 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 406 1 ")" }{TEXT -1 19 " symmetric matrix:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 84 "`A+transp(A)` := evalm(A + transpose(A)) : \+ A + Transp(A) = matrix(`A+transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"-%'TranspG6#F%F&-%'matrixG6#7$7$,$&%\"aG6#\"#6\"\" #,&&F16#\"#7F&&F16#\"#@F&7$F5,$&F16#\"#AF4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation ma y be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "matrix(A) + Transp(matrix(A)) = matrix(`A+tran sp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'matrixG6#7$7$&%\"aG 6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A\"\"\"-%'TranspG6#F%F8-F&6#7$7$,$F* \"\"#,&F.F8F2F87$FB,$F5FA" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 264 "" 0 "" {TEXT 291 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 4 "N.B." }{TEXT -1 60 " The difference of a square matrix and its transpose is a " } {TEXT 293 14 "skew-symmetric" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exempla rily, consider a " }{TEXT 407 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 408 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 409 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 295 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 17 "The difference [" }{TEXT 296 1 "A" }{TEXT -1 1 "]" } {TEXT 410 9 " \226 Transp" }{TEXT -1 1 "[" }{TEXT 297 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 411 1 "(" }{XPPEDIT 18 0 "2" "6#\"\" #" }{TEXT 412 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 413 1 ") " }{TEXT -1 24 " skew-symmetric matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`A-transp(A)` := evalm(A - transpose(A)) : A - Tran sp(A) = matrix(`A-transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,& %\"AG\"\"\"-%'TranspG6#F%!\"\"-%'matrixG6#7$7$\"\"!,&&%\"aG6#\"#7F&&F3 6#\"#@F*7$,&F6F&F2F*F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "matrix(A) - Transp(matrix(A)) = matrix(`A-transp(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\" #77$&F+6#\"#@&F+6#\"#A\"\"\"-%'TranspG6#F%!\"\"-F&6#7$7$\"\"!,&F.F8F2F <7$,&F2F8F.F " 0 "" {MPLTEXT 1 0 65 "A := matrix(3, 3, [a , b, c, d, e, f, g, h, i]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG7 %%\"gG%\"hG%\"iG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "(a) Compute the symmetric part of [" } {TEXT 551 1 "A" }{TEXT -1 7 "] as \275" }{TEXT 558 1 "(" }{TEXT -1 1 "[" }{TEXT 556 1 "A" }{TEXT -1 2 "] " }{TEXT 559 1 "+" }{TEXT -1 1 " \+ " }{TEXT 561 6 "Transp" }{TEXT -1 1 "[" }{TEXT 557 1 "A" }{TEXT -1 1 " ]" }{TEXT 560 1 ")" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "sym(A) :=evalm(1/2*(A + transpose(A))) : Sym(A) = m atrix(sym(A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SymG6#%\"AG-%'m atrixG6#7%7%%\"aG,&%\"bG#\"\"\"\"\"#*&F0F1%\"dGF1F1,&%\"cGF0*&F0F1%\"g GF1F17%F.%\"eG,&%\"fGF0*&F0F1%\"hGF1F17%F5F;%\"iG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "(b) Comput e the antisymmetric part of [" }{TEXT 552 1 "A" }{TEXT -1 7 "] as \+ \275" }{TEXT 564 1 "(" }{TEXT -1 1 "[" }{TEXT 562 1 "A" }{TEXT -1 2 "] " }{TEXT 565 1 "\226" }{TEXT -1 1 " " }{TEXT 567 6 "Transp" }{TEXT -1 1 "[" }{TEXT 563 1 "A" }{TEXT -1 1 "]" }{TEXT 566 1 ")" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "antisym(A) := evalm (1/2*(A - transpose(A))) : Antisym(A) = matrix(antisym(A)) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(AntisymG6#%\"AG-%'matrixG6#7%7%\" \"!,&%\"bG#\"\"\"\"\"#*&#F1F2F1%\"dGF1!\"\",&%\"cGF0*&#F1F2F1%\"gGF1F6 7%,&F5F0*&#F1F2F1F/F1F6F-,&%\"fGF0*&#F1F2F1%\"hGF1F67%,&F;F0*&#F1F2F1F 8F1F6,&FDF0*&#F1F2F1FAF1F6F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "(c) The sum of the two parts of [" }{TEXT 553 1 "A" }{TEXT -1 19 "] yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "`sym(A)+antisym(A)` := evalm(sym(A) + antisym(A)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Sym(A) \+ + Antisym(A) = matrix(`sym(A)+antisym(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$SymG6#%\"AG\"\"\"-%(AntisymGF'F)-%'matrixG6#7%7%% \"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG7%%\"gG%\"hG%\"iG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "which is t he original square matrix [" }{TEXT 554 1 "A" }{TEXT -1 2 "]." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 288 "" 0 "" {TEXT 555 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 301 4 "N.B." }{TEXT -1 20 " Transposition is \+ " }{TEXT 302 9 "reflexive" }{TEXT -1 64 ", i.e. the transpose of a ma trix transpose is the matrix itself" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 334 13 "Transp(Transp" }{TEXT -1 1 "[" }{TEXT 299 1 "A" }{TEXT -1 1 "]" }{TEXT 414 4 ") = " }{TEXT -1 1 "[" }{TEXT 300 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 415 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 416 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 417 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 303 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " A := matrix(3, 2, [ a[11], a[12], a[21], a[22], a[31], a[32] ]) : A \+ = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7 $&%\"aG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A7$&F+6#\"#J&F+6#\"#K" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The transpose of the transpose of the matrix, " }{TEXT 335 13 "Transp(Transp" }{TEXT -1 1 "[" }{TEXT 328 1 "A" }{TEXT -1 1 "]" } {TEXT 418 1 ")" }{TEXT -1 21 ", is the following " }{TEXT 419 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 420 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 421 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 100 "`transp(transp(A))` := transpose(transpose( A)) : Transp(Transp(A)) = matrix(`transp(transp(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#-F%6#%\"AG-%'matrixG6#7%7$&%\"aG6# \"#6&F06#\"#77$&F06#\"#@&F06#\"#A7$&F06#\"#J&F06#\"#K" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "T his operation may be displayed in \"like-in-a-book\" form, viz." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Transp(Transp(matrix(A))) = \+ matrix(`transp(transp(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'T ranspG6#-F%6#-%'matrixG6#7%7$&%\"aG6#\"#6&F/6#\"#77$&F/6#\"#@&F/6#\"#A 7$&F/6#\"#J&F/6#\"#KF)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 267 "" 0 "" {TEXT 304 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 305 4 "N.B." }{TEXT -1 84 " The transpose of the sum of two matrices is the sum of the tw o transposed matrices" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 336 7 "Transp (" }{TEXT -1 1 "[" }{TEXT 306 1 "A" }{TEXT -1 1 "]" }{TEXT 422 3 " + \+ " }{TEXT -1 1 "[" }{TEXT 307 1 "B" }{TEXT -1 1 "]" }{TEXT 423 10 ") = \+ Transp" }{TEXT -1 1 "[" }{TEXT 308 1 "A" }{TEXT -1 1 "]" }{TEXT 424 9 " + Transp" }{TEXT -1 1 "[" }{TEXT 309 1 "B" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Exemplarily, consider " }{TEXT 425 1 "(" }{XPPEDIT 18 0 "2" "6 #\"\"#" }{TEXT 426 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 427 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 310 1 "A" }{TEXT -1 7 "] and \+ [" }{TEXT 311 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A := matrix(2, 3, [a[11], a[12], a[13], a[21], a [22], a[23]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B := mat rix(2, 3, [b[11], b[12], b[13], b[21], b[22], b[23]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{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#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7%&%\"bG6#\"#6&F+6#\"#7&F+6#\"#87%& F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "(a) The transpose of the sum of t he two matrices, " }{TEXT 337 7 "Transp(" }{TEXT -1 1 "[" }{TEXT 324 1 "A" }{TEXT -1 1 "]" }{TEXT 428 3 " + " }{TEXT -1 1 "[" }{TEXT 325 1 "B" }{TEXT -1 1 "]" }{TEXT 429 1 ")" }{TEXT -1 21 ", is the following " }{TEXT 430 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 431 3 " \327 \+ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 432 1 ")" }{TEXT -1 9 " matrix: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`transp(A+B)` := transp ose(evalm(A + B)) : Transp(A + B) = matrix(`transp(A+B)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#,&%\"AG\"\"\"%\"BGF)-%'ma trixG6#7%7$,&&%\"aG6#\"#6F)&%\"bGF3F),&&F26#\"#@F)&F6F9F)7$,&&F26#\"#7 F)&F6F?F),&&F26#\"#AF)&F6FDF)7$,&&F26#\"#8F)&F6FJF),&&F26#\"#BF)&F6FOF )" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "(b) The sum of the two transposed matrices, " }{TEXT 338 6 "Transp" }{TEXT -1 1 "[" }{TEXT 326 1 "A" }{TEXT -1 1 "]" } {TEXT 433 9 " + Transp" }{TEXT -1 1 "[" }{TEXT 327 1 "B" }{TEXT -1 22 "], is the following " }{TEXT 434 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$ " }{TEXT 435 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 436 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "` transp(A)+transp(B)` := evalm(transpose(A) + transpose(B)) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Transp(A) + Transp(B) = matr ix(`transp(A)+transp(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'T ranspG6#%\"AG\"\"\"-F&6#%\"BGF)-%'matrixG6#7%7$,&&%\"aG6#\"#6F)&%\"bGF 5F),&&F46#\"#@F)&F8F;F)7$,&&F46#\"#7F)&F8FAF),&&F46#\"#AF)&F8FFF)7$,&& F46#\"#8F)&F8FLF),&&F46#\"#BF)&F8FQF)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation ma y be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "Transp(matrix(A) + matrix(B)) = matrix(`transp (A)+transp(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#,&-% 'matrixG6#7$7%&%\"aG6#\"#6&F.6#\"#7&F.6#\"#87%&F.6#\"#@&F.6#\"#A&F.6# \"#B\"\"\"-F)6#7$7%&%\"bGF/&FGF2&FGF57%&FGF9&FGF<&FGF?FA-F)6#7%7$,&F-F AFFFA,&F8FAFKFA7$,&F1FAFHFA,&F;FAFLFA7$,&F4FAFIFA,&F>FAFMFA" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 312 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 4 "N.B." }{TEXT -1 117 " The transpose of t he product of two matrices is the product of the two transposed matric es but in the reverse order" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 339 7 " Transp(" }{TEXT -1 1 "[" }{TEXT 314 1 "A" }{TEXT -1 3 "] [" }{TEXT 315 1 "B" }{TEXT -1 1 "]" }{TEXT 437 10 ") = Transp" }{TEXT -1 1 "[" } {TEXT 316 1 "B" }{TEXT -1 2 "] " }{TEXT 340 6 "Transp" }{TEXT -1 1 "[ " }{TEXT 317 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a \+ " }{TEXT 438 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 439 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 440 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 318 1 "A" }{TEXT -1 9 "] and a " }{TEXT 441 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 442 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 443 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 319 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A := matr ix(2, 3, [a[11], a[12], a[13], a[21], a[22], a[23]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B := matrix(3, 2, [b[11], b[12], b[ 21], b[22], b[31], b[32]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@& F+6#\"#A&F+6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6 #7%7$&%\"bG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A7$&F+6#\"#J&F+6#\"#K" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "(a) The transpose of the product of the two matrices, " } {TEXT 341 7 "Transp(" }{TEXT -1 1 "[" }{TEXT 320 1 "A" }{TEXT -1 3 "] \+ [" }{TEXT 321 1 "B" }{TEXT -1 1 "]" }{TEXT 444 1 ")" }{TEXT -1 21 ", \+ is the following " }{TEXT 445 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 446 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 447 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`t ransp(AB)` := transpose(multiply(A, B)) : Transp(A * B) = matrix(`tr ansp(AB)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#*&%\"AG\" \"\"%\"BGF)-%'matrixG6#7$7$,(*&&%\"aG6#\"#6F)&%\"bGF4F)F)*&&F36#\"#7F) &F76#\"#@F)F)*&&F36#\"#8F)&F76#\"#JF)F),(*&&F3F=F)F6F)F)*&&F36#\"#AF)F " 0 "" {MPLTEXT 1 0 63 "`transp(B) transp(A)` := multiply(transpose(B), transpose(A)) :" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Transp(B) * Transp(A) = ma trix(`transp(B) transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-% 'TranspG6#%\"BG\"\"\"-F&6#%\"AGF)-%'matrixG6#7$7$,(*&&%\"aG6#\"#6F)&% \"bGF6F)F)*&&F56#\"#7F)&F96#\"#@F)F)*&&F56#\"#8F)&F96#\"#JF)F),(*&&F5F ?F)F8F)F)*&&F56#\"#AF)F>F)F)*&&F56#\"#BF)FEF)F)7$,(*&F4F)&F9F " 0 "" {MPLTEXT 1 0 61 "Transp(matrix(A) *matrix(B)) = matrix(`transp(B) transp(A)`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#*&-%'matrixG6#7$7%&%\"aG6#\"#6&F.6#\"#7&F. 6#\"#87%&F.6#\"#@&F.6#\"#A&F.6#\"#B\"\"\"-F)6#7%7$&%\"bGF/&FGF27$&FGF9 &FGF<7$&FG6#\"#J&FG6#\"#KFA-F)6#7$7$,(*&F-FAFFFAFA*&F1FAFJFAFA*&F4FAFM FAFA,(*&F8FAFFFAFA*&F;FAFJFAFA*&F>FAFMFAFA7$,(*&F-FAFHFAFA*&F1FAFKFAFA *&F4FAFPFAFA,(*&F8FAFHFAFA*&F;FAFKFAFA*&F>FAFPFAFA" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 287 "" 0 "" {TEXT 541 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (11) for \"" }{TEXT 457 27 "The determina nt of a matrix" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 271 "" 0 "" {TEXT 456 67 "-------------------------------------------------------------- -----" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }