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{PARA 280 "" 0 "" {TEXT 414 38 "MATRICES AND MATRIX OPE RATIONS: Unit 7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 281 "" 0 "" {TEXT 416 23 "Dr. Wlodzislaw Kostecki" }}{PARA 282 "" 0 "" {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }} {PARA 283 "" 0 "" {TEXT -1 54 "Department of Electrical and Communicat ion Engineering" }}{PARA 284 "" 0 "" {TEXT -1 20 "Lae, Morobe Province " }}{PARA 285 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 " " {TEXT -1 0 "" }}{PARA 286 "" 0 "" {TEXT 415 41 "Copyright \251 2000 by Wlodzislaw Kostecki" }}{PARA 287 "" 0 "" {TEXT 417 19 "All rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 288 "" 0 "" {TEXT 418 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 3 "(7)" }{TEXT 348 1 " " }{TEXT 347 25 "Special types of m atrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 541 10 "OBJECTIVES" }{TEXT 542 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 543 1 "\225" } {TEXT -1 88 " To define the five special types of matrices that can b e automatically generated with " }{TEXT 545 5 "Maple" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 544 1 "\225" }{TEXT -1 67 " To provid e alternative methods of inputting the special matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 547 1 "\225" }{TEXT -1 33 " To introduce the co ncept of a " }{TEXT 546 6 "scalar" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 550 1 "\225" }{TEXT -1 35 " To introduce the co ncept of the " }{TEXT 548 11 "unit matrix" }{TEXT -1 6 " or " } {TEXT 549 15 "identity matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 554 1 "\225" }{TEXT -1 33 " To introduce the concept of a \+ " }{TEXT 552 8 "singular" }{TEXT -1 6 " or " }{TEXT 553 14 "non-inve rtible" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 555 1 "\225" }{TEXT -1 38 " To introduce the concept of matrix " }{TEXT 556 8 "sparsity" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 551 1 "\225" }{TEXT -1 62 " To specify some properties of the special typ es of matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "restart : with(linalg, coldim, de t, diag, eigenvals, inverse, randmatrix, rowdim, transpose) :" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Normally, when defining and inputting a matrix, it is necessary to specify and enter " }{TEXT 278 5 "every" }{TEXT -1 24 " element \+ of the matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "There are some kinds of matrices that may be generated with " }{TEXT 513 5 "Maple" }{TEXT -1 94 " 'automaticall y'. All these special types of matrices are included and discussed in \+ this Unit." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 273 "" 0 "" {TEXT 307 1 "A" }{TEXT -1 2 ". " }{TEXT 277 19 "The diagon al matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 5 "The " }{TEXT 262 15 "diagonal matrix" }{TEXT -1 7 " is a " }{TEXT 258 13 "square matrix" }{TEXT -1 14 " that has non-" }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 11 " elements " }{TEXT 376 4 "only " }{TEXT -1 9 " on its " }{TEXT 259 9 "principal" }{TEXT -1 4 " ( " }{TEXT 260 4 "main" }{TEXT -1 3 ", " }{TEXT 261 7 "leading" }{TEXT -1 77 " ) diagonal, which runs from the top left of the matrix to the bottom right." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 303 5 "Maple" }{TEXT -1 12 ", there are \+ " }{TEXT 328 3 "two" }{TEXT -1 65 " alternative methods of defining an d inputting diagonal matrices." }}{PARA 2 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 377 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 378 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 379 1 ")" }{TEXT -1 19 " diagonal matrix [" }{TEXT 276 1 "A" } {TEXT -1 2 "]." }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 357 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 358 4 " diag" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "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 360 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 359 8 "diagonal" }{TEXT -1 60 " ind exing function at either of the two locations under the " }{TEXT 370 5 "array" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "A := array(diagonal, 1..4, 1..4, [(1,1)=a[11], (2,2)= 0, (3,3)=a[33], (4,4)=a[44]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A := array(1..4, 1..4, [(1,1)=a[11], (2,2)=0, (3,3)=a[33], (4 ,4)=a[44]], diagonal) : 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 263 4 "N.B." }{TEXT -1 62 " Some of the elements in the principal diagonal may also be " }{XPPMATH 20 "6#%&zerosG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "For example, substituting arb itrary numerical values for the non-" }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 26 " elements of the matrix [" }{TEXT 265 1 "A" }{TEXT -1 7 "] giv es" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := subs(a[11]=3, a[ 33]=2, a[44]=5, matrix(A)) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'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 267 "" 0 "" {TEXT 319 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 4 "N.B." }{TEXT -1 5 " A \+ " }{TEXT 443 8 "diagonal" }{TEXT -1 20 " matrix, in which " }{TEXT 267 3 "all" }{TEXT -1 29 " the diagonal elements are " }{TEXT 320 5 "equal" }{TEXT -1 15 " is called a " }{TEXT 266 6 "scalar" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "For example, construct an " }{TEXT 436 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 437 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 438 1 ")" }{TEXT -1 17 " scalar matrix [" } {TEXT 435 1 "A" }{TEXT -1 22 "] whose elements are " }{XPPEDIT 18 0 " d" "6#%\"dG" }{TEXT -1 8 ". Let " }{XPPEDIT 18 0 "n=4" "6#/%\"nG\"\" %" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "This may be done using either of the pre ceding Methods 1 or 2. Where large diagonal matrices are involved, the method using the double " }{TEXT 439 3 "for" }{TEXT -1 24 "-loop cons truct and the " }{TEXT 440 8 "diagonal" }{TEXT -1 49 " indexing functi on could be more convenient, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n := 4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := array(diagonal, 1..n, 1..n) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for i to n do for j to n do if j = i then A[i,j] := d fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := matrix(A) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&%\"dG\"\"!F+F+7&F+F*F+F+7&F+F+F*F+7&F+F+F+ F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "i := 'i' :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 266 "" 0 "" {TEXT 318 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 4 "N.B." } {TEXT -1 5 " A " }{TEXT 444 6 "scalar" }{TEXT -1 21 " matrix, in w hich " }{TEXT 269 3 "all" }{TEXT -1 29 " the diagonal elements are \+ " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 17 " is called the " }{TEXT 271 11 "unit matrix" }{TEXT -1 6 " or " }{TEXT 272 15 "identity matr ix" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " }{TEXT 442 11 "unit matrix" }{TEXT -1 23 ", refer to Unit (9). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 265 "" 0 "" {TEXT 317 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 4 "N.B." }{TEXT -1 5 " A " }{TEXT 445 8 "diagonal" }{TEXT -1 12 " matrix is " }{TEXT 274 3 "not" }{TEXT -1 13 " changed by " }{TEXT 275 13 "transposition" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " } {TEXT 441 9 "transpose" }{TEXT -1 36 " of a matrix, refer to Unit (10 ). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT 316 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 321 4 "N.B." }{TEXT -1 7 " Any " } {TEXT 322 6 "square" }{TEXT -1 30 " matrix can be reduced to a " } {TEXT 446 8 "diagonal" }{TEXT -1 20 " matrix having non-" }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 11 " elements " }{TEXT 323 4 "only" }{TEXT -1 27 " in the principal diagonal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 324 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 308 1 "B" }{TEXT -1 2 ". " }{TEXT 279 20 "The symmetric matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Th e " }{TEXT 280 16 "symmetric matrix" }{TEXT -1 7 " is a " }{TEXT 281 6 "square" }{TEXT -1 23 " matrix, in which the " }{TEXT 380 2 "( \+ " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT 381 2 ", " }{XPPEDIT 18 0 "j" "6 #%\"jG" }{TEXT 382 2 " )" }{TEXT -1 28 "th element is equal to the " }{TEXT 383 2 "( " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT 385 2 ", " } {XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT 384 2 " )" }{TEXT -1 99 "th element , so that the pattern of the matrix elements has a symmetry about the \+ principal diagonal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 300 5 "Maple" }{TEXT -1 12 ", there are " }{TEXT 329 3 "two" }{TEXT -1 66 " alternative met hods of defining and inputting symmetric matrices." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider \+ a " }{TEXT 386 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 387 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 388 1 ")" }{TEXT -1 20 " symme tric matrix [" }{TEXT 282 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 362 8 "Method 1" } {TEXT -1 12 ". Using the " }{TEXT 361 9 "symmetric" }{TEXT -1 60 " ind exing function at either of the two locations under the " }{TEXT 371 5 "array" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "A := array(symmetric, 1..3, 1..3, [(1,1)=5, (2,2)=2, \+ (3,3)=7, (1,2)=1, (1,3)=3, (2,3)=-2]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "A:=array(1..3, 1..3, [(1,1)=5, (2,2)=2, (3,3)=7, (1, 2)=1, (1,3)=3, (2,3)=-2], symmetric) : 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 363 8 "Method 2" }{TEXT -1 51 ". Using any of the meth ods found in Unit (1), e.g.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A := matrix(3, 3, [5, 1, 3, 1, 2, -2, 3, -2, 7]) : A = matrix( A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"&\" \"\"\"\"$7%F+\"\"#!\"#7%F,F/\"\"(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 294 "" 0 "" {TEXT 519 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 520 4 "N.B." } {TEXT -1 105 " Raising a symmetric matrix to any (positive or negativ e) integer power yields another symmetric matrix." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "For exampl e, compute powers " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" }{TEXT -1 23 " of the above ma trix [" }{TEXT 521 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "`A^2` := evalm(A^2) : `A^(-3)` := evalm(A^(-3)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A^2 = matrix(`A^2`) ; A ^ ` -3` = matrix(`A^(-3)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)% \"AG\"\"#\"\"\"-%'matrixG6#7%7%\"#NF(\"#M7%F(\"\"*!#:7%F/F2\"#i" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-3G-%'matrixG6#7%7%#\"&ML\" \"%(>##!%)z\"\"$p\"#!&*)H\"F.7%F/#\"$V#\"#8#\"%` " 0 "" {MPLTEXT 1 0 72 "B := matrix(3, 3, \+ [1, a, a^2, a, a^2, 1, a^2, 1, a]) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"\"%\"aG*$)F+\"\"#F*7%F+F ,F*7%F,F*F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 315 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 4 "N.B." }{TEXT -1 5 " A " } {TEXT 447 9 "symmetric" }{TEXT -1 12 " matrix is " }{TEXT 285 3 "not " }{TEXT -1 13 " changed by " }{TEXT 286 13 "transposition" }{TEXT -1 118 ". Based on this property, some sources define the symmetric m atrix as a matrix, which is equal to its transpose, viz." }}}{EXCHG {PARA 290 "" 0 "" {TEXT -1 1 "[" }{TEXT 424 1 "A" }{TEXT -1 2 "] " } {TEXT 423 1 "=" }{TEXT -1 1 " " }{TEXT 426 6 "Transp" }{TEXT -1 1 "[" }{TEXT 425 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 451 18 "Pre-multiplication" }{TEXT -1 41 " of either side of this formula by the " }{TEXT 452 7 "invers e" }{TEXT -1 6 " of [" }{TEXT 449 1 "A" }{TEXT -1 21 "] yields the id entity" }}}{EXCHG {PARA 292 "" 0 "" {TEXT 456 4 "(Inv" }{TEXT -1 1 "[ " }{TEXT 453 1 "A" }{TEXT -1 1 "]" }{TEXT 457 1 ")" }{TEXT -1 1 " " } {TEXT 458 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 454 1 "A" }{TEXT -1 1 "] " }{TEXT 459 1 ")" }{TEXT -1 1 " " }{TEXT 460 1 "=" }{TEXT -1 2 " [" } {TEXT 455 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where [" }{TEXT 461 1 "U" } {TEXT -1 10 "] is the " }{TEXT 462 4 "unit" }{TEXT -1 30 " matrix of the same size as [" }{TEXT 463 1 "A" }{TEXT -1 92 "]. Some sources de fine the symmetric matrix as a matrix, which satisfies the above ident ity." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "[ For the " }{TEXT 450 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 22 "Notice that the sum [" } {TEXT 465 1 "A" }{TEXT -1 2 "] " }{TEXT 464 1 "+" }{TEXT -1 1 " " } {TEXT 467 6 "Transp" }{TEXT -1 1 "[" }{TEXT 466 1 "A" }{TEXT -1 30 "] \+ is also a symmetric matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " } {TEXT 469 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 471 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 470 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 468 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 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 10 "The sum [" }{TEXT 473 1 "A" }{TEXT -1 2 "] " }{TEXT 472 1 "+" }{TEXT -1 1 " \+ " }{TEXT 475 6 "Transp" }{TEXT -1 1 "[" }{TEXT 474 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 476 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 478 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 477 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`A +transp(A)` := evalm(A + transpose(A)) : A + Transp(A) = matrix(`A+t ransp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"-%'Trans pG6#F%F&-%'matrixG6#7%7%,$%\"aG\"\"#,&%\"bGF&%\"dGF&,&%\"cGF&%\"gGF&7% F2,$%\"eGF1,&%\"fGF&%\"hGF&7%F5F;,$%\"iGF1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "which is symmetri c \226 see also Unit (10) for an interesting use of this property." }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 325 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 326 4 "N.B." }{TEXT -1 5 " A " }{TEXT 448 9 "s ymmetric" }{TEXT -1 30 " matrix can be reduced to a " }{TEXT 327 8 " diagonal" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 314 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 275 "" 0 "" {TEXT 309 1 "C" } {TEXT -1 2 ". " }{TEXT 264 41 "The skew-symmetric (antisymmetric) matr ix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The skew-symmetric (antisymmetric) matrix is a " }{TEXT 287 6 "square" }{TEXT -1 23 " matrix, in which the " }{TEXT 392 2 "( \+ " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT 393 2 ", " }{XPPEDIT 18 0 "j" "6 #%\"jG" }{TEXT 394 2 " )" }{TEXT -1 35 "th element is the negative of \+ the " }{TEXT 395 2 "( " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT 397 2 ", \+ " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT 396 2 " )" }{TEXT -1 21 "th elem ent, so that " }{TEXT 288 3 "all" }{TEXT -1 51 " the principal-diago nal elements are necessarily " }{XPPMATH 20 "6#%&zerosG" }{TEXT -1 106 ", whilst the pattern of elements has a symmetry about the princi pal diagonal but with a reversal of sign." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 301 5 "Maple" }{TEXT -1 12 ", there are " }{TEXT 330 3 "two" }{TEXT -1 71 " \+ alternative methods of defining and inputting skew-symmetric matrices. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 398 1 "(" }{XPPEDIT 18 0 "3" "6#\"\" $" }{TEXT 399 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 400 1 ") " }{TEXT -1 25 " skew-symmetric matrix [" }{TEXT 289 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 365 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 364 13 "a ntisymmetric" }{TEXT -1 60 " indexing function at either of the two lo cations under the " }{TEXT 372 5 "array" }{TEXT -1 10 " function:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A := array(antisymmetric, 1. .3, 1..3, [(1,2)=1, (1,3)=5, (2,3)=-3]) :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 87 "A := array(1..3, 1..3, [(1,2)=1, (1,3)=5, (2,3)=-3] , antisymmetric) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"AG-%'matrixG6#7%7%\"\"!\"\"\"\"\"&7%!\"\"F*!\"$7%!\"&\"\"$F*" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 366 8 "Method 2" }{TEXT -1 51 ". Using any of the methods found in Uni t (1), e.g.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A := matrix (3, 3, [0, 1, 5, -1, 0, -3, -5, 3, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"!\"\"\"\"\"&7%!\" \"F*!\"$7%!\"&\"\"$F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 296 "" 0 "" {TEXT 523 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 524 4 "N.B." }{TEXT -1 41 " Raising an antisymmetric matrix to an " }{TEXT 525 4 "even" }{TEXT -1 26 " integer power yields a " }{TEXT 526 9 "symmetric" } {TEXT -1 19 " matrix, while an " }{TEXT 527 3 "odd" }{TEXT -1 27 " i nteger power yields an " }{TEXT 528 13 "antisymmetric" }{TEXT -1 9 " \+ matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "For example, compute powers " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 23 " of the above matrix [" }{TEXT 529 1 "A" }{TEXT -1 2 "]." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "`A^2` := evalm(A^2) : `A^3 ` := evalm(A^3) : A^2 = matrix(`A^2`) ; A^3 = matrix(`A^3`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"#\"\"\"-%'matrixG6#7%7%!# E\"#:!\"$7%F/!#5!\"&7%F0F3!#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)% \"AG\"\"$\"\"\"-%'matrixG6#7%7%\"\"!!#N!$v\"7%\"#NF.\"$0\"7%\"$v\"!$0 \"F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 5 "The " }{TEXT 530 6 "square" }{TEXT -1 6 " of [" } {TEXT 531 1 "A" }{TEXT -1 6 "] is " }{TEXT 532 9 "symmetric" }{TEXT -1 12 ", but the " }{TEXT 533 4 "cube" }{TEXT -1 6 " of [" }{TEXT 534 1 "A" }{TEXT -1 6 "] is " }{TEXT 535 13 "antisymmetric" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 297 "" 0 "" {TEXT 536 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Another example of a sk ew-symmetric matrix is the following " }{TEXT 401 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 402 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 403 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 290 1 "B" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "B := matrix(3, 3, \+ [0, a, b, -a, 0, c, -b, -c, 0]) : B = matrix(B) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"!%\"aG%\"bG7%,$F+!\"\"F*%\" cG7%,$F,F/,$F0F/F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 313 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 4 "N.B." }{TEXT -1 43 " A n antisymmetric matrix changes sign in " }{TEXT 292 13 "transposition " }{TEXT -1 158 " but is otherwise unchanged. Based on this property, some sources define the antisymmetric matrix as a matrix, which is eq ual to its negative transpose, viz." }}}{EXCHG {PARA 291 "" 0 "" {TEXT -1 1 "[" }{TEXT 428 1 "A" }{TEXT -1 2 "] " }{TEXT 427 1 "=" } {TEXT -1 1 " " }{TEXT 430 1 "\226" }{TEXT -1 1 " " }{TEXT 483 6 "Trans p" }{TEXT -1 1 "[" }{TEXT 429 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Notice th at the difference [" }{TEXT 479 1 "A" }{TEXT -1 2 "] " }{TEXT 482 1 " \226" }{TEXT -1 1 " " }{TEXT 481 6 "Transp" }{TEXT -1 1 "[" }{TEXT 480 1 "A" }{TEXT -1 35 "] is also an antisymmetric matrix." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 485 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 487 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 486 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 484 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 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 17 "The difference [" }{TEXT 488 1 "A" } {TEXT -1 2 "] " }{TEXT 494 1 "\226" }{TEXT -1 1 " " }{TEXT 490 6 "Tran sp" }{TEXT -1 1 "[" }{TEXT 489 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 491 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 493 3 " \327 \+ " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 492 1 ")" }{TEXT -1 9 " 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%!\"\"-%'ma trixG6#7%7%\"\"!,&%\"bGF&%\"dGF*,&%\"cGF&%\"gGF*7%,&F3F&F2F*F0,&%\"fGF &%\"hGF*7%,&F6F&F5F*,&F;F&F:F*F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "which is antisymmetric \+ \226 see also Unit (10) for an interesting use of this property." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Notice the result of summation of [" }{TEXT 496 1 "A" }{TEXT -1 2 "] " }{TEXT 495 1 "+" }{TEXT -1 1 " " }{TEXT 498 6 "Transp" } {TEXT -1 1 "[" }{TEXT 497 1 "A" }{TEXT -1 9 "] and [" }{TEXT 499 1 " A" }{TEXT -1 2 "] " }{TEXT 502 1 "\226" }{TEXT -1 1 " " }{TEXT 501 6 " Transp" }{TEXT -1 1 "[" }{TEXT 500 1 "A" }{TEXT -1 7 "], viz." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "`A+transp(A) + A-transp(A)` \+ := evalm((A + transpose(A)) + (A - transpose(A))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "[A + Transp(A)] * ` + ` * [A - Transp(A)] = matrix(`A+transp(A) + A-transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(7#,&%\"AG\"\"\"-%'TranspG6#F'F(F(%$~+~GF(7#,&F'F(F)!\"\"F(-%' matrixG6#7%7%,$%\"aG\"\"#,$%\"bGF7,$%\"cGF77%,$%\"dGF7,$%\"eGF7,$%\"fG F77%,$%\"gGF7,$%\"hGF7,$%\"iGF7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "which is " }{XPPEDIT 18 0 " 2" "6#\"\"#" }{TEXT -1 2 " [" }{TEXT 503 1 "A" }{TEXT -1 53 "] \226 se e Unit (10) for the application of this result." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 312 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 305 4 "N.B." }{TEXT -1 34 " The skew-symmetric matrix is a " }{TEXT 306 8 "singular" }{TEXT -1 6 " or " }{TEXT 510 14 "non-invertible" } {TEXT -1 35 " matrix, i.e. its determinant is " }{XPPMATH 20 "6#%%ze roG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "For example, the determinant of th e above antisymmetrix matrix [" }{TEXT 506 1 "A" }{TEXT -1 2 "] " } {TEXT 509 1 "\226" }{TEXT -1 1 " " }{TEXT 508 6 "Transp" }{TEXT -1 1 " [" }{TEXT 507 1 "A" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`det(A-transp(A))` := det(`A-transp(A)`) : Det(A - \+ Transp(A)) = `det(A-transp(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$DetG6#,&%\"AG\"\"\"-%'TranspG6#F(!\"\"\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "[ Refer to U nit (11) for the " }{TEXT 504 11 "determinant" }{TEXT -1 28 " and to Unit (14) for the " }{TEXT 505 7 "inverse" }{TEXT -1 23 " of a squa re matrix. ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 311 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 276 "" 0 "" {TEXT 310 1 "D" }{TEXT -1 2 ". " }{TEXT 293 17 "The sparse matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 294 6 "sparse" } {TEXT -1 69 " matrix is a rectangular matrix, in which most of the el ements are " }{XPPMATH 20 "6#%&zerosG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 302 5 "Maple" }{TEXT -1 12 ", there are " }{TEXT 331 3 "two " }{TEXT -1 63 " alternative methods of defining and inputting sparse \+ matrices." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 12 "Consider a " }{TEXT 404 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 405 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 406 1 ")" }{TEXT -1 17 " sparse matrix [" }{TEXT 295 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 368 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 367 6 "sp arse" }{TEXT -1 60 " indexing function at either of the two locations \+ under the " }{TEXT 373 5 "array" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A := array(sparse, 1..3, 1..4, [(1, 3)=2, (2,1)=-3, (2,4)=4, (3,2)=-1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "A := array(1..3, 1..4, [(1,3)=2, (2,1)=-3, (2,4)=4, ( 3,2)=-1], sparse) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7&\"\"!F*\"\"#F*7&!\"$F*F*\"\"%7&F*!\"\"F*F*" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 369 8 "Method 2" }{TEXT -1 51 ". Using any of the methods found \+ in Unit (1), e.g.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "A := \+ matrix(3, 4, [0, 0, 2, 0, -3, 0, 0, 4, 0, -1, 0, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7&\"\"!F*\" \"#F*7&!\"$F*F*\"\"%7&F*!\"\"F*F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Another example of a spars e matrix is the following " }{TEXT 407 1 "(" }{XPPEDIT 18 0 "6" "6#\" \"'" }{TEXT 408 3 " \327 " }{XPPEDIT 18 0 "6" "6#\"\"'" }{TEXT 409 1 " )" }{TEXT -1 10 " matrix [" }{TEXT 350 1 "Y" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "Y := array(sparse, 1..6, 1. .6, [(1,1)=15, (1,2)=-10, (1,3)=-5, (2,1)=-10, (2,2)=10, (3,1)=-5, (3, 3)=25, (3,4)=-12, (3,5)=-8, (4,3)=-12, (4,4)=12, (5,3)=-8, (5,5)=23, ( 5,6)=-15, (6,5)=-15, (6,6)=15]) : Y = matrix(Y) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"YG-%'matrixG6#7(7(\"#:!#5!\"&\"\"!F-F-7(F+\"#5F-F -F-F-7(F,F-\"#D!#7!\")F-7(F-F-F2\"#7F-F-7(F-F-F3F-\"#B!#:7(F-F-F-F-F8F *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 278 "" 0 " " {TEXT 374 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 349 4 "N.B." }{TEXT -1 14 " The matrix [ " }{TEXT 351 1 "Y" }{TEXT -1 134 "] was obtained when analysing electr ic power flow in a real-life six bus-bar transmission system. Multipli cation of every element of [" }{TEXT 421 1 "Y" }{TEXT -1 44 "] by the \+ imaginary unit yields a so-called " }{TEXT 410 21 "bus admittance mat rix" }{TEXT -1 16 " of the system." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "It is typical of bus ad mittance matrices that the vast majority of their elements have the va lue " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 40 ". Such matrices are sai d to be highly " }{TEXT 352 6 "sparse" }{TEXT -1 18 ". The notion of " }{TEXT 353 15 "matrix sparsity" }{TEXT -1 52 " is used to determi ne how sparse such matrices are." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Since realistic-size syste ms may have in excess of " }{XPPEDIT 18 0 "100" "6#\"$+\"" }{TEXT -1 174 " bus-bars, it is convenient to have a method for fast determinat ion of the sparsity of bus admittance matrices. A suitable method is p roposed below, where the above matrix [" }{TEXT 354 1 "Y" }{TEXT -1 24 "] is used as an example." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "(a) Find the number of all elem ents of the matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "dim( Y) := rowdim(Y) * coldim(Y) : all_elements(Y) = dim(Y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-all_elementsG6#%\"YG\"#O" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "( b) Find the number of " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 31 "-value d elements of the matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "n := 0 : for i to rowdim(Y) do for j to coldim(Y) do if Y[i, j] = 0 then n := n+1 fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "zeros(Y) := n : zero_elements(Y) = zeros(Y) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.zero_elementsG6#%\"YG\"#?" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "(c) Compute the matrix sparsity according to the formula" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "sparsity(Y) = zero_elements( Y)/all_elements(Y) ; sparsity(Y) = evalf(zeros(Y)/dim(Y), 3) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)sparsityG6#%\"YG*&-%.zero_elements GF&\"\"\"-%-all_elementsGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %)sparsityG6#%\"YG$\"$c&!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 279 "" 0 "" {TEXT 375 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 4 "N.B." } {TEXT -1 26 " This particular matrix [" }{TEXT 355 1 "Y" }{TEXT -1 6 "] is " }{TEXT 297 8 "singular" }{TEXT -1 13 " since its " }{TEXT 298 11 "determinant" }{TEXT -1 6 " is " }{XPPMATH 20 "6#%%zeroG" } {TEXT -1 33 ", as evidenced by the following:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "`det(Y)` := det(Y) : Det(Y) = `det(Y)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"YG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "This im plies that matrix [" }{TEXT 356 1 "Y" }{TEXT -1 10 "] has no " } {TEXT 299 7 "inverse" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 304 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 277 "" 0 "" {TEXT 333 1 "E" }{TEXT -1 2 ". " }{TEXT 332 21 "The unimodular matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Th e " }{TEXT 334 10 "unimodular" }{TEXT -1 14 " matrix is a " }{TEXT 346 6 "square" }{TEXT -1 2 " " }{TEXT 512 7 "integer" }{TEXT -1 19 " \+ matrix that has " }{TEXT 511 3 "all" }{TEXT -1 47 " elements on its principal diagonal equal to " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 46 ". The determinant of a unimodular matrix is " }{XPPMATH 20 "6#%& unityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 335 5 "Maple" }{TEXT -1 155 ", a unimodular matrix may be defined and input using any of th e methods found in Unit (1) or it can be generated by the program. To \+ this end, the function " }{TEXT 336 10 "randmatrix" }{TEXT -1 29 " tog ether with the type name " }{TEXT 337 10 "unimodular" }{TEXT -1 10 " a re used." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 411 1 "(" } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 412 3 " \327 " }{XPPEDIT 18 0 "4" " 6#\"\"%" }{TEXT 413 1 ")" }{TEXT -1 21 " unimodular matrix [" }{TEXT 338 1 "A" }{TEXT -1 27 "] generated by the program." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A := randmatrix(4, 4, unimodular) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7& \"\"\"!#&)!#b!#P7&\"\"!F*!#N\"#(*7&F/F/F*\"#]7&F/F/F/F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "T he determinant of [" }{TEXT 339 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `de t(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 342 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 340 4 "N.B." }{TEXT -1 7 " The " }{TEXT 341 7 "inverse" }{TEXT -1 53 " of a unimodular matrix is also a unimodular \+ matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The inverse of the above matrix [" }{TEXT 431 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv(A )` := inverse(A) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"AG-%'matrixG6#7&7&\"\"\"\"#&)\"%II!'3(f\"7 &\"\"!F-\"#N!%Z=7&F2F2F-!#]7&F2F2F2F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The determinant o f the inverse of [" }{TEXT 432 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`det(inv(A))` := det(`inv(A)`) : \+ Det(Inv(A)) = `det(inv(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$D etG6#-%$InvG6#%\"AG\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 271 "" 0 "" {TEXT 343 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 344 4 "N.B." }{TEXT -1 2 " " }{TEXT 345 13 "Transposition" }{TEXT -1 160 " of a unimodul ar matrix returns a unimodular matrix whose elements are a mirror refl ection of the elements of the original matrix about the principal diag onal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The transpose of the above matrix [" }{TEXT 433 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`trans p(A)` := transpose(A) : Transp(A) = matrix(`transp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#%\"AG-%'matrixG6#7&7&\"\"\"\"\" !F.F.7&!#&)F-F.F.7&!#b!#NF-F.7&!#P\"#(*\"#]F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The determinan t of the transpose of [" }{TEXT 434 1 "A" }{TEXT -1 4 "] is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`det(transp(A))` := det(`tra nsp(A)`) : Det(Transp(A)) = `det(transp(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#-%'TranspG6#%\"AG\"\"\"" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 293 "" 0 "" {TEXT 514 5 "* * *" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 537 4 "N.B." }{TEXT -1 107 " Raising a unimodular matrix to any (positive or negative) integer power yields another unimodular matrix ." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "For example, compute powers " }{XPPEDIT 18 0 "2" "6#\"\" #" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" }{TEXT -1 23 " of the above matrix [" }{TEXT 538 1 "A" }{TEXT -1 2 "]." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "`A^2` := evalm(A^2) : `A^( -3)` := evalm(A^(-3)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " A^2 = matrix(`A^2`) ; A ^ ` -3` = matrix(`A^(-3)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"#\"\"\"-%'matrixG6#7&7&F(!$q\"\"%lG!& p5\"7&\"\"!F(!#q!%c:7&F2F2F(\"$+\"7&F2F2F2F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%$~-3G-%'matrixG6#7&7&\"\"\"\"$b#\"&:!=!(fLb\"7& \"\"!F,\"$0\"!&\"z57&F1F1F,!$]\"7&F1F1F1F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Both resultant ma trices are unimodular." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 298 "" 0 "" {TEXT 539 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 515 4 "N.B." }{TEXT -1 11 " All the " }{TEXT 516 11 "eigenvalues" }{TEXT -1 30 " of a u nimodular matrix are " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The eigenvalues of the above unimodular matrix [" }{TEXT 517 1 "A" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "c harroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6&\"\"\"F)F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " }{TEXT 518 11 "eigenvalues" }{TEXT -1 36 " \+ of a matrix, refer to Unit (21). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 289 "" 0 "" {TEXT 422 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "P roceed to Unit (8) for \"" }{TEXT 420 38 "Multiplication of a matrix b y a number" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 272 "" 0 "" {TEXT 419 67 "------------------------------------------------------------------ -" }}}}{MARK "281 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }