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"Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 418 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 419 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(17)" }{TEXT 340 1 " " }{TEXT 339 21 "The orthogonal ma trix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 458 10 "OBJECTIVES" }{TEXT 459 1 ":" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 460 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 461 10 "orthogonal" }{TEXT -1 9 " ma trix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 462 1 "\225" }{TEXT -1 28 " To introduce the function " }{TEXT 463 6 "orthog" }{TEXT -1 40 " for tes ting matrices for orthogonality." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 466 1 "\225" }{TEXT -1 82 " To provide examples of orthogonal matrices wi th numerical and symbolic elements." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 465 1 "\225" }{TEXT -1 64 " To specify and illustrate properties of t he orthogonal matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 464 1 "\225" } {TEXT -1 80 " To investigate properties of certain operations involvi ng orthogonal matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "restart : with(linalg, det , exponential, inverse, multiply, orthog, transpose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "If the " }{TEXT 261 7 "inverse" }{TEXT -1 7 " of a " }{TEXT 260 4 "rea l" }{TEXT -1 32 " square matrix is equal to its " }{TEXT 262 9 "trans pose" }{TEXT -1 34 ", then the matrix is said to be " }{TEXT 259 10 "orthogonal" }{TEXT -1 19 ". Thus, a matrix [" }{TEXT 258 1 "A" } {TEXT -1 18 "] is orthogonal if" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 317 3 "Inv" }{TEXT -1 1 "[" }{TEXT 263 1 "A" }{TEXT -1 1 "]" }{TEXT 347 9 " = Transp" }{TEXT -1 1 "[" }{TEXT 264 1 "A" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The determinant of an orthogonal matrix is " }{XPPEDIT 18 0 "1 " "6#\"\"\"" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\" " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 348 1 " (" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 349 3 " \327 " }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 350 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 265 1 " A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "A := matrix(3, 3, [6/7, 2/7, 3/7, 3/7, -6/7, -2/7, 2/7, 3/7, -6/7] ) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matr ixG6#7%7%#\"\"'\"\"(#\"\"#F,#\"\"$F,7%F/#!\"'F,#!\"#F,7%F-F/F2" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "A " }{TEXT 467 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 18 " returned by the " }{TEXT 346 6 "orthog " }{TEXT -1 54 " function verifies that the matrix is orthogonal, viz. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "orthog(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "(a) The inverse of the mat rix, " }{TEXT 343 3 "Inv" }{TEXT -1 1 "[" }{TEXT 342 1 "A" }{TEXT -1 22 "], is the following " }{TEXT 351 1 "(" }{XPPEDIT 18 0 "3" "6#\" \"$" }{TEXT 352 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 353 1 " )" }{TEXT -1 9 " matrix:" }}}{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/#\"\"#F/7%F2#!\"'F/F07%F0#!\"#F/F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "(b) The transpose of the matrix, " }{TEXT 345 6 "Transp" }{TEXT -1 1 "[" }{TEXT 344 1 "A" }{TEXT -1 22 "], is the following " }{TEXT 354 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 355 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 356 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`transp(A)` := transpose(A) : Transp(A) = matrix(`t ransp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#%\"AG-%'m atrixG6#7%7%#\"\"'\"\"(#\"\"$F/#\"\"#F/7%F2#!\"'F/F07%F0#!\"#F/F5" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Both matrices of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The determin ant of [" }{TEXT 425 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "A s another example, consider a " }{TEXT 357 1 "(" }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 358 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 359 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 341 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "A := matrix(2, 2 , [cos(phi), sin(phi), -sin(phi), cos(phi)]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$-%$cosG6#%$phiG- %$sinGF,7$,$F.!\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "(a) The inverse of the matrix, " }{TEXT 318 3 "Inv" }{TEXT -1 1 "[" }{TEXT 266 1 "A" }{TEXT -1 22 "], \+ is the following " }{TEXT 360 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 361 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 362 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`i nv(A)` := combine(inverse(A)) : Inv(A) = matrix(`inv(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"AG-%'matrixG6#7$7$*&-%$co sG6#%$phiG\"\"\",&*$)F.\"\"#F2F2*$)-%$sinGF0F6F2F2!\"\",$*&F9F2F3F;F;7 $F=F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Simplification yields the following form of the above \+ matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "`inv(A)` := map( simplify, `inv(A)`) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$InvG6#%\"AG-%'matrixG6#7$7$-%$cosG6#%$phiG,$-%$si nGF/!\"\"7$F2F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "(b) The transpose of the matrix, " } {TEXT 319 6 "Transp" }{TEXT -1 1 "[" }{TEXT 267 1 "A" }{TEXT -1 22 "], is the following " }{TEXT 363 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 364 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 365 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`t ransp(A)` := transpose(A) : Transp(A) = matrix(`transp(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#%\"AG-%'matrixG6#7$7$-%$c osG6#%$phiG,$-%$sinGF/!\"\"7$F2F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Both matrices of (a) and ( b) are equal, so matrix [" }{TEXT 424 1 "A" }{TEXT -1 34 "] is orthogo nal. A test using the " }{TEXT 423 6 "orthog" }{TEXT -1 29 " function \+ verifies this, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "orth og(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Let, in t he above matrix [" }{TEXT 268 1 "A" }{TEXT -1 4 "], " }{XPPEDIT 18 0 "phi=-3*Pi/2" "6#/%$phiG,$*(\"\"$\"\"\"%#PiGF(\"\"#!\"\"F+" }{TEXT -1 38 " to obtain a numerical example. Thus," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "A := subs(phi=-3*Pi/2, matrix(A)) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$-%$cosG6#, $%#PiG#!\"$\"\"#-%$sinGF,7$,$F2!\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Evaluation of [" }{TEXT 338 1 "A" }{TEXT -1 34 "] is performed using the function " } {TEXT 468 3 "map" }{TEXT -1 49 " and the arrow-type procedure includin g function " }{TEXT 469 4 "eval" }{TEXT -1 6 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A := map(x->eval(x), A) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"!\"\" \"7$!\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "(a) The inverse of the matrix, " }{TEXT 335 3 "Inv" }{TEXT -1 1 "[" }{TEXT 334 1 "A" }{TEXT -1 22 "], is the follow ing " }{TEXT 366 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 367 3 " \+ \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 368 1 ")" }{TEXT -1 9 " ma trix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv(A)` := invers e(A) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$InvG6#%\"AG-%'matrixG6#7$7$\"\"!!\"\"7$\"\"\"F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "( b) The transpose of the matrix, " }{TEXT 337 6 "Transp" }{TEXT -1 1 " [" }{TEXT 336 1 "A" }{TEXT -1 22 "], is the following " }{TEXT 369 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 370 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 371 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`transp(A)` := transpose(A) : Tra nsp(A) = matrix(`transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%' TranspG6#%\"AG-%'matrixG6#7$7$\"\"!!\"\"7$\"\"\"F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Both matr ices of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The determinant of [" }{TEXT 426 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 269 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 4 "N .B." }{TEXT -1 7 " The " }{TEXT 457 7 "product" }{TEXT -1 10 " of t he " }{TEXT 273 9 "transpose" }{TEXT -1 9 " of an " }{TEXT 280 10 " orthogonal" }{TEXT -1 19 " matrix and the " }{TEXT 274 6 "matrix" } {TEXT -1 20 " itself obeys the " }{TEXT 271 15 "commutative law" } {TEXT -1 50 ". The product matrix is the appropriately sized " } {TEXT 272 4 "unit" }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 320 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 275 1 "A" }{TEXT -1 1 "] " }{TEXT 372 1 ")" }{TEXT -1 2 " [" }{TEXT 276 1 "A" }{TEXT -1 1 "]" } {TEXT 373 3 " = " }{TEXT -1 1 "[" }{TEXT 277 1 "A" }{TEXT -1 2 "] " } {TEXT 374 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 278 1 "A" }{TEXT -1 1 "] " }{TEXT 375 4 ") = " }{TEXT -1 1 "[" }{TEXT 279 1 "U" }{TEXT -1 1 "] " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "As an example, consider the above matrix [" }{TEXT 281 1 "A" }{TEXT -1 26 "] with numerical elements." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(a) The product " }{TEXT 376 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 321 1 "A" }{TEXT -1 1 "]" }{TEXT 377 1 ")" }{TEXT -1 2 " [" }{TEXT 322 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 378 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 379 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 380 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`t ransp(A) A` := multiply(transpose(A), A) : Transp(A)*A = matrix(`tra nsp(A) A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'TranspG6#%\"AG\" \"\"F(F)-%'matrixG6#7$7$F)\"\"!7$F/F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product \+ [" }{TEXT 323 1 "A" }{TEXT -1 2 "] " }{TEXT 381 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 324 1 "A" }{TEXT -1 1 "]" }{TEXT 382 1 ")" }{TEXT -1 20 " is the following " }{TEXT 383 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 384 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 385 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`A transp(A)` := multiply(A, transpose(A)) : A* `Transp(A)` = matrix(` A transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%*Tra nsp(A)GF&-%'matrixG6#7$7$F&\"\"!7$F-F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Both product matr ices are equal to the " }{TEXT 386 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"# " }{TEXT 387 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 388 1 ")" }{TEXT -1 15 " unit matrix [" }{TEXT 282 1 "U" }{TEXT -1 2 "]." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 283 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 4 "N.B." }{TEXT -1 7 " The " }{TEXT 284 4 "unit" }{TEXT -1 16 " matrix is an " }{TEXT 285 10 "orthogonal" } {TEXT -1 14 " matrix, i.e." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 325 3 " Inv" }{TEXT -1 1 "[" }{TEXT 288 1 "U" }{TEXT -1 1 "]" }{TEXT 389 9 " = Transp" }{TEXT -1 1 "[" }{TEXT 289 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "A s an example, consider the " }{TEXT 390 1 "(" }{XPPEDIT 18 0 "3" "6# \"\"$" }{TEXT 391 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 392 1 ")" }{TEXT -1 15 " unit matrix [" }{TEXT 287 1 "U" }{TEXT -1 2 "]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "U := array(1..3, 1..3, \+ identity) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"U G-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "(a) The inve rse of the matrix, " }{TEXT 327 3 "Inv" }{TEXT -1 1 "[" }{TEXT 326 1 "U" }{TEXT -1 6 "], is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " `inv(U)` := inverse(U) : Inv(U) = matrix(`inv(U)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"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 34 "(b) The transpose of the matrix, " }{TEXT 329 6 "Transp" }{TEXT -1 1 "[" }{TEXT 328 1 "U" }{TEXT -1 6 "], is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`transp(U)` := transpose(U) \+ : Transp(U) = matrix(`transp(U)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#%\"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 51 "Both matrices of (a) and (b) are equal, so matrix [" }{TEXT 428 1 "U" }{TEXT -1 34 "] is orthogonal. A test using the " }{TEXT 427 6 "orthog" }{TEXT -1 29 " function verifies this, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "orthog(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 258 "" 0 "" {TEXT 290 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 4 "N.B." }{TEXT -1 7 " The " }{TEXT 292 7 "inverse" }{TEXT -1 9 " of an " }{TEXT 452 10 "orthogonal" }{TEXT -1 21 " matrix is also an " }{TEXT 453 10 "orthogonal" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "As a numerical ex ample, consider the " }{TEXT 393 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 394 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 395 1 ")" } {TEXT -1 21 " orthogonal matrix [" }{TEXT 293 1 "A" }{TEXT -1 33 "] u sed earlier in this Unit, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := matrix(2, 2, [0, 1, -1, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"!\"\"\"7$!\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Let the inverse of this matrix be named [" }{TEXT 294 1 " B" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B := \+ inverse(A) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" BG-%'matrixG6#7$7$\"\"!!\"\"7$\"\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "(a) The inverse o f the matrix, " }{TEXT 331 3 "Inv" }{TEXT -1 1 "[" }{TEXT 295 1 "B" } {TEXT -1 6 "], is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv( B)` := inverse(B) : Inv(B) = matrix(`inv(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"BG-%'matrixG6#7$7$\"\"!\"\"\"7$!\"\"F-" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "(b) The transpose of the matrix, " }{TEXT 330 6 "Transp" } {TEXT -1 1 "[" }{TEXT 296 1 "B" }{TEXT -1 6 "], is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`transp(B)` := transpose(B) : Transp(B) = matrix(`transp(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Transp G6#%\"BG-%'matrixG6#7$7$\"\"!\"\"\"7$!\"\"F-" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Since " }{TEXT 332 3 "Inv" }{TEXT -1 1 "[" }{TEXT 297 1 "B" }{TEXT -1 1 "]" }{TEXT 396 9 " = Transp" }{TEXT -1 1 "[" }{TEXT 298 1 "B" }{TEXT -1 16 "], t he matrix [" }{TEXT 299 1 "B" }{TEXT -1 44 "] is an orthogonal matrix. A test using the " }{TEXT 429 6 "orthog" }{TEXT -1 29 " function veri fies this, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "orthog(B ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 300 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 301 4 "N.B." }{TEXT -1 7 " The " }{TEXT 454 7 "product" }{TEXT -1 6 " of " }{TEXT 455 10 "orthogonal" }{TEXT -1 41 " matrices of the sa me order is also an " }{TEXT 456 10 "orthogonal" }{TEXT -1 28 " matr ix. This requires that" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 397 8 "\{Tra nsp(" }{TEXT -1 1 "[" }{TEXT 302 1 "A" }{TEXT -1 3 "] [" }{TEXT 303 1 "B" }{TEXT -1 1 "]" }{TEXT 398 2 ")\}" }{TEXT -1 1 " " }{TEXT 400 1 " \{" }{TEXT -1 1 "[" }{TEXT 304 1 "A" }{TEXT -1 3 "] [" }{TEXT 305 1 "B " }{TEXT -1 1 "]" }{TEXT 399 4 "\} = " }{TEXT -1 1 "[" }{TEXT 306 1 "U " }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Exemplarily, consider the two orthogonal \+ matrices [" }{TEXT 307 1 "A" }{TEXT -1 7 "] and [" }{TEXT 308 1 "B" } {TEXT -1 19 "] used before, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "A := matrix(2, 2, [0, 1, -1, 0]) : B := matrix(2, \+ 2, [0, -1, 1, 0]) : 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 19 "(a) The transpose " }{TEXT 333 7 "Transp(" }{TEXT -1 1 "[" }{TEXT 309 1 "A" }{TEXT -1 3 "] [" }{TEXT 310 1 "B" }{TEXT -1 1 "]" }{TEXT 401 1 ")" }{TEXT -1 20 " is the following " }{TEXT 402 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 403 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 404 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`transp(AB)` := transpose(multiply( A, B)) : Transp(A * B) = matrix(`transp(AB)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#*&%\"AG\"\"\"%\"BGF)-%'matrixG6#7$7$F)\"\" !7$F0F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product [" }{TEXT 311 1 "A" }{TEXT -1 3 "] \+ [" }{TEXT 312 1 "B" }{TEXT -1 21 "] is the following " }{TEXT 405 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 406 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 407 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`AB` := multiply(A, B) : A * B = \+ matrix(`AB`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"BGF &-%'matrixG6#7$7$F&\"\"!7$F-F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(c) The product " }{TEXT 408 8 "\{Transp(" }{TEXT -1 1 "[" }{TEXT 313 1 "A" }{TEXT -1 3 "] [" } {TEXT 314 1 "B" }{TEXT -1 1 "]" }{TEXT 409 2 ")\}" }{TEXT -1 1 " " } {TEXT 411 1 "\{" }{TEXT -1 1 "[" }{TEXT 315 1 "A" }{TEXT -1 3 "] [" } {TEXT 316 1 "B" }{TEXT -1 1 "]" }{TEXT 410 1 "\}" }{TEXT -1 20 " is t he following " }{TEXT 412 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 413 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 414 1 ")" }{TEXT -1 14 " unit matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "`t ransp(AB) AB`:=multiply(`transp(AB)`, `AB`) : Transp(A*B) * A*B = matr ix(`transp(AB) AB`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%'TranspG 6#*&%\"AG\"\"\"%\"BGF*F*F)F*F+F*-%'matrixG6#7$7$F*\"\"!7$F1F*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 430 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "As an example of an orthogonal matrix who se determinant is " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 27 ", consider the following " }{TEXT 432 1 "(" }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 433 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 434 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 431 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "A := matrix(3, 3, [2/3, - 2/3, 1/3, 1/3, 2/3, 2/3, 2/3, 1/3, -2/3]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%#\"\"#\"\"$#!\"# F,#\"\"\"F,7%F/F*F*7%F*F/F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "A test for orthogonality of [" } {TEXT 435 1 "A" }{TEXT -1 7 "] gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "orthog(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The determinant of [" }{TEXT 436 1 "A" }{TEXT -1 4 "] is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : \+ Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG !\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 437 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 438 4 "N.B." }{TEXT -1 7 " The " } {TEXT 448 11 "exponential" }{TEXT -1 18 " function of an " }{TEXT 449 13 "antisymmetric" }{TEXT -1 16 " matrix is an " }{TEXT 450 10 " orthogonal" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 440 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 441 3 " \327 \+ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 442 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 439 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A := matrix(2, 2, [0, -alpha, alpha, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$ \"\"!,$%&alphaG!\"\"7$F,F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The exponential function of [" } {TEXT 443 1 "A" }{TEXT -1 30 "] yields the following matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "`exp(A)` := exponential(A) \+ : exp(A) = matrix(`exp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ expG6#%\"AG-%'matrixG6#7$7$-%$cosG6#%&alphaG,$-%$sinGF/!\"\"7$F2F-" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "A test for orthogonality of " }{TEXT 446 4 "exp(" }{TEXT -1 1 "[" }{TEXT 445 1 "A" }{TEXT -1 1 "]" }{TEXT 447 1 ")" }{TEXT -1 7 " g ives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "orthog(`exp(A)`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " } {TEXT 451 20 "exponential function" }{TEXT -1 32 " of matrices, refer to Section " }{TEXT 444 1 "A" }{TEXT -1 16 " of Unit (23). ]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 422 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (18) for \"" }{TEXT 421 51 "Replacing a column in a matrix with a column matrix" }{TEXT -1 2 " \"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 420 67 "----------------------- --------------------------------------------" }}}}{MARK "9 0" 0 } {VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }