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1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Ari al Narrow" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 876 39 "MATRICES AND MATRIX OPE RATIONS: Unit 16" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 258 "" 0 "" {TEXT 878 23 "Dr. Wlodzislaw Kostecki" }}{PARA 260 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 260 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 260 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 260 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 877 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 879 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 880 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(16)" }{TEXT 581 1 " " }{TEXT 580 36 "The complex matri x and its conjugate" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1113 10 "OBJECTIVES" }{TEXT 1114 1 ":" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1115 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 1116 16 "complex -numbered" }{TEXT -1 6 " or " }{TEXT 1188 7 "complex" }{TEXT -1 9 " \+ matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1130 1 "\225" }{TEXT -1 162 " To provide alternative methods of creating matrices whose elements \+ are real parts, imaginary parts, moduli, and arguments of elements of \+ a given complex matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1147 1 "\225 " }{TEXT -1 86 " To provide alternative methods of creating matrices \+ whose elements are expressed in " }{TEXT 1148 5 "Maple" }{TEXT 1150 1 "\222" }{TEXT -1 3 "s " }{TEXT 1149 10 "polar form" }{TEXT -1 1 "." } }}{EXCHG {PARA 0 "" 0 "" {TEXT 1155 1 "\225" }{TEXT -1 140 " To provi de alternative methods of creating matrices whose elements are express ed in the special polar form used in electrical engineering." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1158 1 "\225" }{TEXT -1 28 " To introduc e the function " }{TEXT 1156 5 "evalc" }{TEXT -1 36 " for evaluation o f complex matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1157 1 "\225" } {TEXT -1 183 " To investigate the methods of display and evaluation o f complex matrices whose elements are given in the polar or other tran scendental form containing decimal or irrational numbers." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1159 1 "\225" }{TEXT -1 65 " To provide an exam ple of a matrix comprising functions of the " }{TEXT 1160 14 "imagina ry unit" }{TEXT -1 25 ", which evaluates to a " }{TEXT 1161 4 "real " }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1162 1 "\225 " }{TEXT -1 20 " To introduce the " }{TEXT 1163 8 "k-matrix" }{TEXT -1 33 " and investigate its properties." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1191 1 "\225" }{TEXT -1 85 " To provide a universal method for \+ presenting the product of complex matrices with " }{TEXT 1194 8 "symb olic" }{TEXT -1 46 " elements as the sum of a matrix comprising " } {TEXT 1192 4 "real" }{TEXT -1 33 " parts and a matrix comprising " } {TEXT 1193 9 "imaginary" }{TEXT -1 42 " parts of elements of the prod uct matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1176 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 1177 9 "conjugate" }{TEXT -1 22 " of a \+ complex matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1178 1 "\225" }{TEXT -1 91 " To provide alternative methods of computation of the conjugat e of a given complex matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1180 1 " \225" }{TEXT -1 20 " To introduce the " }{TEXT 1179 9 "Hermitian" } {TEXT -1 40 " matrix and investigate its properties." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1181 1 "\225" }{TEXT -1 65 " To investigate pro perties of the conjugate of a complex matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1182 1 "\225" }{TEXT -1 100 " To investigate properties of c ertain operations involving both a complex matrix and its conjugate." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1187 1 "\225" }{TEXT -1 57 " To inves tigate properties of operations involving the " }{TEXT 1183 11 "deter minant" }{TEXT -1 31 " of a complex matrix and the " }{TEXT 1184 11 "determinant" }{TEXT -1 26 " of the matrix conjugate." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1186 1 "\225" }{TEXT -1 35 " To introduce the c oncept of the " }{TEXT 1185 21 "imaginary unit matrix" }{TEXT -1 33 " and investigate its properties." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "restart : with(lin alg, adj, coldim, det, diag, inverse, multiply, rowdim, transpose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 287 1 "A" }{TEXT -1 2 ". " }{TEXT 288 18 "The complex matrix" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 1189 16 "complex-numbered" }{TEXT -1 6 " or " } {TEXT 1190 7 "complex" }{TEXT -1 40 " matrix is a matrix, in which at least " }{TEXT 556 3 "one" }{TEXT -1 15 " element is a " }{TEXT 259 14 "complex number" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 689 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 279 3 " \327 \+ " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 690 1 ")" }{TEXT -1 18 " comple x matrix [" }{TEXT 280 1 "Z" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "Z := matrix(3, 3, [1+5*I, 2+6*I, 3 -7*I, 4+8*I, 5-9*I, 6+10*I, 7+11*I, 8+12*I, 9-13*I]) : Z = matrix(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"ZG-%'matrixG6#7%7%^$\"\"\"\" \"&^$\"\"#\"\"'^$\"\"$!\"(7%^$\"\"%\"\")^$F,!\"*^$F/\"#57%^$\"\"(\"#6^ $F6\"#7^$\"\"*!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The various operations on the complex mat rix [" }{TEXT 915 1 "Z" }{TEXT -1 25 "] are analysed hereunder." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 445 1 "\225" }{TEXT -1 32 " To obtain a matrix containing " }{TEXT 282 4 "only" }{TEXT -1 6 " the " }{TEXT 283 4 "real" }{TEXT -1 27 " \+ part of the elements of [" }{TEXT 281 1 "Z" }{TEXT -1 48 "], use any o f the following alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 583 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 584 5 "evalm" }{TEXT -1 5 " and " }{TEXT 585 2 " Re" }{TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`Re(Z)` := evalm(Re(Z)) : `Re(Z)` = matrix(`Re(Z)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Re(Z)G-%'matrixG6#7%7%\"\"\"\"\"#\" \"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"\"*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 586 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 587 3 "map" }{TEXT -1 5 " and " }{TEXT 588 2 "Re" }{TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "`Re(Z)` := map(Re, Z) : `Re(Z)` = matrix(`Re(Z)`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Re(Z)G-%'matrixG6#7%7%\"\"\"\" \"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"\"*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 917 8 "Method 3" }{TEXT -1 12 ". Using the " }{TEXT 918 3 "map" }{TEXT -1 5 " and " }{TEXT 919 2 "op" }{TEXT -1 50 " functions together with the arrow-type proce dure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`Re(Z)` := map(x-> op(1, x), Z) : `Re(Z)` = matrix(`Re(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Re(Z)G-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\" \"'7%\"\"(\"\")\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 446 1 "\225" }{TEXT -1 32 " To obtain a \+ matrix containing " }{TEXT 285 4 "only" }{TEXT -1 6 " the " }{TEXT 569 15 "numerical value" }{TEXT -1 10 " of the " }{TEXT 286 9 "imagi nary" }{TEXT -1 27 " part of the elements of [" }{TEXT 284 1 "Z" } {TEXT -1 48 "], use any of the following alternative methods." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 589 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 590 5 "evalm" } {TEXT -1 5 " and " }{TEXT 591 2 "Im" }{TEXT -1 11 " functions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`Im(Z)` := evalm(Im(Z)) : \+ `Im(Z)` = matrix(`Im(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Im(Z )G-%'matrixG6#7%7%\"\"&\"\"'!\"(7%\"\")!\"*\"#57%\"#6\"#7!#8" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 592 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 593 3 "map" } {TEXT -1 5 " and " }{TEXT 594 2 "Im" }{TEXT -1 11 " functions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "`Im(Z)` := map(Im, Z) : `I m(Z)` = matrix(`Im(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Im(Z)G -%'matrixG6#7%7%\"\"&\"\"'!\"(7%\"\")!\"*\"#57%\"#6\"#7!#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 923 8 "M ethod 3" }{TEXT -1 21 ". Using the function " }{TEXT 924 3 "map" } {TEXT -1 24 " and twice the function " }{TEXT 925 2 "op" }{TEXT -1 40 " together with the arrow-type procedure:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "`Im(Z)` := map(x->op(1, op(2, x)), Z) : `Im(Z)` = matrix(`Im(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Im(Z)G-%'matr ixG6#7%7%\"\"&\"\"'!\"(7%\"\")!\"*\"#57%\"#6\"#7!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 568 1 "\225" } {TEXT -1 32 " To obtain a matrix containing " }{TEXT 566 4 "only" } {TEXT -1 6 " the " }{TEXT 567 9 "imaginary" }{TEXT -1 27 " part of t he elements of [" }{TEXT 565 1 "Z" }{TEXT -1 48 "], use any of the fol lowing alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 595 8 "Method 1" }{TEXT -1 12 ". Using \+ the " }{TEXT 596 5 "evalm" }{TEXT -1 5 " and " }{TEXT 597 2 "Re" } {TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "`Im(Z) I` := evalm(Z - `Re(Z)`) : `Im(Z) I` = matrix(`Im(Z) I`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(Im(Z)~IG-%'matrixG6#7%7%^#\"\"&^# \"\"'^#!\"(7%^#\"\")^#!\"*^#\"#57%^#\"#6^#\"#7^#!#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 920 8 "Method \+ 2" }{TEXT -1 12 ". Using the " }{TEXT 921 3 "map" }{TEXT -1 5 " and " }{TEXT 922 2 "op" }{TEXT -1 50 " functions together with the arrow-typ e procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "`Im(Z) I` : = map(x->op(2, x), Z) : `Im(Z) I` = matrix(`Im(Z) I`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(Im(Z)~IG-%'matrixG6#7%7%\"\"&\"\"'!\"(7%\"\" )!\"*\"#57%\"#6\"#7!#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 598 8 "Method 3" }{TEXT -1 22 ". Using th e functions " }{TEXT 599 5 "evalm" }{TEXT -1 5 " and " }{TEXT 600 2 "I m" }{TEXT -1 34 " multiplied by the imaginary unit:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`Im(Z) I` := evalm(Im(Z) * I) : `Im(Z) \+ I` = matrix(`Im(Z) I`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(Im(Z)~I G-%'matrixG6#7%7%^#\"\"&^#\"\"'^#!\"(7%^#\"\")^#!\"*^#\"#57%^#\"#6^#\" #7^#!#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 601 8 "Method 4" }{TEXT -1 22 ". Using the functions " } {TEXT 602 3 "map" }{TEXT -1 5 " and " }{TEXT 603 2 "Im" }{TEXT -1 34 " multiplied by the imaginary unit:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`Im(Z) I` := map(Im * I, Z) : `Im(Z) I` = matrix(`I m(Z) I`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(Im(Z)~IG-%'matrixG6#7 %7%^#\"\"&^#\"\"'^#!\"(7%^#\"\")^#!\"*^#\"#57%^#\"#6^#\"#7^#!#8" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 557 1 "\225" }{TEXT -1 43 " To obtain a matrix with the elements of [ " }{TEXT 558 1 "Z" }{TEXT -1 15 "] expressed in " }{TEXT 559 5 "Maple " }{TEXT 916 1 "\222" }{TEXT -1 3 "s " }{TEXT 560 10 "polar form" } {TEXT -1 7 ", i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "z[i, j] = polar(z[i,j]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"zG6$%\"iG %\"jG-%&polarG6$-%$absG6#F$-%)argumentGF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "use any of the fo llowing alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 604 8 "Method 1" }{TEXT -1 21 ". Using the function " }{TEXT 606 3 "map" }{TEXT -1 14 " and function " } {TEXT 607 7 "convert" }{TEXT -1 53 " together with the arrow-type proc edure and function " }{TEXT 605 5 "polar" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Z_p := map(x->convert(x, polar), Z) : Z_p = matrix(Z_p) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$Z_pG-%' matrixG6#7%7%-%&polarG6$*$-%%sqrtG6#\"#E\"\"\"-%'arctanG6#\"\"&-F+6$,$ *$-F/6#\"#5F2\"\"#-F46#\"\"$-F+6$*$-F/6#\"#eF2,$-F46##\"\"(FA!\"\"7%-F +6$,$*$-F/6#F6F2\"\"%-F46#F>-F+6$*$-F/6#\"$1\"F2,$-F46##\"\"*F6FM-F+6$ ,$*$-F/6#\"#MF2F>-F46##F6FA7%-F+6$*$-F/6#\"$q\"F2-F46##\"#6FL-F+6$,$*$ -F/6#\"#8F2FU-F46##FAF>-F+6$,$F:F6,$-F46##FhpF\\oFM" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 608 8 "Method \+ 2" }{TEXT -1 21 ". Using the function " }{TEXT 609 3 "map" }{TEXT -1 14 " and function " }{TEXT 610 5 "polar" }{TEXT -1 40 " together with \+ the arrow-type procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Z_p := map(x->polar(x), Z) : Z_p = matrix(Z_p) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$Z_pG-%'matrixG6#7%7%-%&polarG6$*$-%%sqrtG6#\"#E \"\"\"-%'arctanG6#\"\"&-F+6$,$*$-F/6#\"#5F2\"\"#-F46#\"\"$-F+6$*$-F/6# \"#eF2,$-F46##\"\"(FA!\"\"7%-F+6$,$*$-F/6#F6F2\"\"%-F46#F>-F+6$*$-F/6# \"$1\"F2,$-F46##\"\"*F6FM-F+6$,$*$-F/6#\"#MF2F>-F46##F6FA7%-F+6$*$-F/6 #\"$q\"F2-F46##\"#6FL-F+6$,$*$-F/6#\"#8F2FU-F46##FAF>-F+6$,$F:F6,$-F46 ##FhpF\\oFM" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 611 8 "Method 3" }{TEXT -1 12 ". Using the " }{TEXT 612 3 "map" }{TEXT -1 5 " and " }{TEXT 613 5 "polar" }{TEXT -1 22 " fu nctions (simplest):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Z_p \+ := map(polar, Z) : Z_p = matrix(Z_p) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$Z_pG-%'matrixG6#7%7%-%&polarG6$*$-%%sqrtG6#\"#E\"\"\"-%'arcta nG6#\"\"&-F+6$,$*$-F/6#\"#5F2\"\"#-F46#\"\"$-F+6$*$-F/6#\"#eF2,$-F46## \"\"(FA!\"\"7%-F+6$,$*$-F/6#F6F2\"\"%-F46#F>-F+6$*$-F/6#\"$1\"F2,$-F46 ##\"\"*F6FM-F+6$,$*$-F/6#\"#MF2F>-F46##F6FA7%-F+6$*$-F/6#\"$q\"F2-F46# #\"#6FL-F+6$,$*$-F/6#\"#8F2FU-F46##FAF>-F+6$,$F:F6,$-F46##FhpF\\oFM" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 563 1 "\225" }{TEXT -1 37 " To obtain a matrix containing the \+ " }{TEXT 562 7 "modulus" }{TEXT -1 50 " (absolute value, magnitude) o f the elements of [" }{TEXT 561 1 "Z" }{TEXT -1 48 "], use any of the \+ following alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 614 8 "Method 1" }{TEXT -1 12 ". Usi ng the " }{TEXT 615 3 "map" }{TEXT -1 2 ", " }{TEXT 616 4 "sqrt" } {TEXT -1 2 ", " }{TEXT 617 2 "Re" }{TEXT -1 6 ", and " }{TEXT 618 2 "I m" }{TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "`abs(Z)` := map(sqrt(Re^2+Im^2), Z) : abs(Z) = matrix(`abs(Z)` ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"ZG-%'matrixG6#7%7% *$-%%sqrtG6#\"#E\"\"\"*$-F/6#\"#SF2*$-F/6#\"#eF27%*$-F/6#\"#!)F2*$-F/6 #\"$1\"F2*$-F/6#\"$O\"F27%*$-F/6#\"$q\"F2*$-F/6#\"$3#F2*$-F/6#\"$]#F2 " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 619 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 620 5 "evalm " }{TEXT -1 5 " and " }{TEXT 621 3 "abs" }{TEXT -1 11 " functions:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`abs(Z)` := evalm(abs(Z)) \+ : abs(Z) = matrix(`abs(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ absG6#%\"ZG-%'matrixG6#7%7%*$-%%sqrtG6#\"#E\"\"\",$*$-F/6#\"#5F2\"\"#* $-F/6#\"#eF27%,$*$-F/6#\"\"&F2\"\"%*$-F/6#\"$1\"F2,$*$-F/6#\"#MF2F87%* $-F/6#\"$q\"F2,$*$-F/6#\"#8F2FC,$F4FB" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 622 8 "Method 3" }{TEXT -1 12 ". Using the " }{TEXT 623 3 "map" }{TEXT -1 5 " and " }{TEXT 624 3 "abs" }{TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "`abs(Z)` := map(abs, Z) : abs(Z) = matrix(`abs(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"ZG-%'matrixG6#7%7% *$-%%sqrtG6#\"#E\"\"\",$*$-F/6#\"#5F2\"\"#*$-F/6#\"#eF27%,$*$-F/6#\"\" &F2\"\"%*$-F/6#\"$1\"F2,$*$-F/6#\"#MF2F87%*$-F/6#\"$q\"F2,$*$-F/6#\"#8 F2FC,$F4FB" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 625 8 "Method 4" }{TEXT -1 21 ". Using the function " } {TEXT 626 3 "map" }{TEXT -1 14 " and function " }{TEXT 627 2 "op" } {TEXT -1 91 " together with the arrow-type procedure applied to the fi rst argument (modulus) of matrix [" }{TEXT 628 3 "Z_p" }{TEXT -1 2 "]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`abs(Z)` := map(x->op(1 , x), Z_p) : abs(Z) = matrix(`abs(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"ZG-%'matrixG6#7%7%*$-%%sqrtG6#\"#E\"\"\",$ *$-F/6#\"#5F2\"\"#*$-F/6#\"#eF27%,$*$-F/6#\"\"&F2\"\"%*$-F/6#\"$1\"F2, $*$-F/6#\"#MF2F87%*$-F/6#\"$q\"F2,$*$-F/6#\"#8F2FC,$F4FB" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "F loating-point evaluation of any of the above " }{TEXT 564 5 "exact" } {TEXT -1 43 " results gives the following approximation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`approx(abs(Z))` := evalf(matrix(`a bs(Z)`)) : abs(Z) = matrix(`approx(abs(Z))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"ZG-%'matrixG6#7%7%$\"+9&>!*4&!\"*$\"+?`bCj F/$\"+1Jx:wF/7%$\"+7>FW*)F/$\"+9IcH5!\")$\"+z.>m6F97%$\"+\"[SQI\"F9$\" +50AU9F9$\"+I)Q6e\"F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 462 1 "\225" }{TEXT -1 37 " To obtain a \+ matrix containing the " }{TEXT 542 18 "principal argument" }{TEXT -1 2 " " }{TEXT 1117 1 "(" }{TEXT -1 4 "in " }{TEXT 1119 7 "radians" } {TEXT 1118 1 ")" }{TEXT -1 22 " of the elements of [" }{TEXT 463 1 "Z " }{TEXT -1 51 "], use either of the following alternative methods." } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 629 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 630 5 "evalm " }{TEXT -1 5 " and " }{TEXT 631 8 "argument" }{TEXT -1 11 " functions :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`arg(Z)` := evalm(argu ment(Z)) : Arg(Z) = matrix(`arg(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ArgG6#%\"ZG-%'matrixG6#7%7%-%'arctanG6#\"\"&-F.6#\"\"$,$-F.6 ##\"\"(F3!\"\"7%-F.6#\"\"#,$-F.6##\"\"*F0F9-F.6##F0F37%-F.6##\"#6F8-F. 6##F3F=,$-F.6##\"#8FBF9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 632 8 "Method 2" }{TEXT -1 12 ". Using th e " }{TEXT 633 3 "map" }{TEXT -1 5 " and " }{TEXT 634 8 "argument" } {TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`arg(Z)` := map(argument, Z) : Arg(Z) = matrix(`arg(Z)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ArgG6#%\"ZG-%'matrixG6#7%7%-%'arct anG6#\"\"&-F.6#\"\"$,$-F.6##\"\"(F3!\"\"7%-F.6#\"\"#,$-F.6##\"\"*F0F9- F.6##F0F37%-F.6##\"#6F8-F.6##F3F=,$-F.6##\"#8FBF9" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 635 8 "Method 3" }{TEXT -1 21 ". Using the function " }{TEXT 636 3 "map" }{TEXT -1 14 " and function " }{TEXT 637 2 "op" }{TEXT -1 114 " together with the ar row-type procedure applied to the second argument (principal argument \+ in radians) of matrix [" }{TEXT 638 3 "Z_p" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`arg(Z)` := map(x->op(2, x), Z_p) : Arg(Z) = matrix(`arg(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ArgG6#%\"ZG-%'matrixG6#7%7%-%'arctanG6#\"\"&-F.6#\"\"$,$-F.6## \"\"(F3!\"\"7%-F.6#\"\"#,$-F.6##\"\"*F0F9-F.6##F0F37%-F.6##\"#6F8-F.6# #F3F=,$-F.6##\"#8FBF9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Floating-point evaluation of any o f the above " }{TEXT 464 5 "exact" }{TEXT -1 42 " result gives the fol lowing approximation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`a pprox(arg(Z))` := evalf(matrix(`arg(Z)`)) : Arg(Z) = matrix(`approx( arg(Z))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ArgG6#%\"ZG-%'matri xG6#7%7%$\"+n2St8!\"*$\"+sd/\\7F/$!+SX!f;\"F/7%$\"+=([r5\"F/$!+Aypj5F/ $\"+FoPI5F/7%$\"+4r1/5F/$\"+Ks$z#)*!#5$!+Im^_'*F@" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1120 1 "\225" } {TEXT -1 37 " To obtain a matrix containing the " }{TEXT 1122 18 "pr incipal argument" }{TEXT -1 2 " " }{TEXT 1123 1 "(" }{TEXT -1 4 "in \+ " }{TEXT 1125 7 "degrees" }{TEXT 1124 1 ")" }{TEXT -1 22 " of the ele ments of [" }{TEXT 1121 1 "Z" }{TEXT -1 48 "], use any of the followin g alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1126 8 "Method 1" }{TEXT -1 12 ". Using t he " }{TEXT 1127 5 "evalm" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "`deg(arg(Z))` := evalf(evalm(`arg(Z)` *18 0/Pi)) : deg(Arg(Z)) = matrix(`deg(arg(Z))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$degG6#-%$ArgG6#%\"ZG-%'matrixG6#7%7%$\"+^n+py!\")$ \"+8^]crF2$!+V49!o'F27%$\"+\")[\\VjF2$!+(eRX4'F2$\"+ZVi.fF27%$\"+o2)Gv &F2$\"+YK*4j&F2$!+WY[IbF2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1128 8 "Method 2" }{TEXT -1 12 ". Using t he " }{TEXT 1129 3 "map" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 92 "`deg(arg(Z))` := evalf(map(x->x*180/Pi, `arg (Z)`)) : deg(Arg(Z)) = matrix(`deg(arg(Z))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$degG6#-%$ArgG6#%\"ZG-%'matrixG6#7%7%$\"+^n+py!\")$ \"+8^]crF2$!+V49!o'F27%$\"+\")[\\VjF2$!+(eRX4'F2$\"+ZVi.fF27%$\"+o2)Gv &F2$\"+YK*4j&F2$!+WY[IbF2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1151 8 "Method 3" }{TEXT -1 21 ". Using t he function " }{TEXT 1152 3 "map" }{TEXT -1 14 " and function " } {TEXT 1153 2 "op" }{TEXT -1 114 " together with the arrow-type procedu re applied to the second argument (principal argument in radians) of m atrix [" }{TEXT 1154 3 "Z_p" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "`deg(arg(Z))` := evalf(evalm(map(x->op(2, x), Z _p) *180/Pi)) : deg(Arg(Z)) = matrix(`deg(arg(Z))`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$degG6#-%$ArgG6#%\"ZG-%'matrixG6#7%7%$\"+^n+p y!\")$\"+8^]crF2$!+V49!o'F27%$\"+\")[\\VjF2$!+(eRX4'F2$\"+ZVi.fF27%$\" +o2)Gv&F2$\"+YK*4j&F2$!+WY[IbF2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 570 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 571 4 "N.B." } {TEXT -1 70 " A customary practice in electrical engineering is to re present the " }{TEXT 892 11 "exponential" }{TEXT -1 135 " form of a \+ complex number using a shorthand notation, which includes a (rounded-o ff) modulus and (rounded-off) principal argument in " }{TEXT 572 7 "d egrees" }{TEXT -1 53 ". This notation as used in textbooks looks like this" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {BITMAP 306 26 26 1 "?TMGWQB:n@>?N`;JGj:>=>::::vyyyyy=::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: J:vYxI:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::;:xI:::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::;:xI:::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::;:xI:::::::::::::::: ::::::::::::::;J:<:::::::::::::::::::::::::::::::::::::yay=::::::::::: :::::::::::::::::::::::::::::::::::<::;::::::::::::::::::::::::::::::: ::vYxI::;jysy::::::::::::::::::::::::::::::::::::::::>:;:yA:>:jy;::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::B:Zy=: ::::J::yA:>:jy;:::::::::::::::::::::::::;jysy::::::::::::::::::::::::: :::::::::::::::>:;:yA:>:jy;::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::B:Zy=:::::J::yA:>:jy;::::::::::::::::::::::: ::;jysy:Z:B:;:vYJ::yA>:jysyZ:J:ryvY::<:;Zy=jy;B:>:xI:ya:J::yay=<:;Zy=j y;;Z::xIyA:>:xuYyuyyyy;:J:B:Zy=<::ryvY:<:;::vYxI:;Z:jy;ry>:<:ryZ::xIB: ZymyC:>:jysyZ:J::yA>:jysy>:<:yay=::::::::::J:B:vYxI:;:vY::::::J:B:Zy=< :ryZ::xIyA>:<:yay=:J:B:vY:xI:;Z:jy;ry>:<:ryvYJ:B:vY:xI;Z::xIyA:Z::xI;Z ::xIB::Zymy;B:>::jysy:>:<:yAZyM:B:Zy=<:ryZ::xIyA>:<:ryvYJ:B:Zy=<:ryZ:j ysy::::::>:yay=:J:>:vYJ::yA>:jy;;:vY<:ry:>:jyC:>:jysy>:jyC:>:jysy>:jy; 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This form is referred to as the " }{TEXT 894 5 "polar" }{TEXT -1 27 " form of a complex number." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "T he symbol of " }{TEXT 895 9 "Kennelly\222" }{TEXT -1 30 "s operator \+ does not exist in " }{TEXT 896 5 "Maple" }{TEXT -1 200 ". Therefore, i t is necessary to develop a method for representing elements of a comp lex matrix in a way resembling the above \"engineering\" polar form. T hree alternative methods are proposed hereunder." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1131 6 "Step 1" } {TEXT -1 24 ". Obtain a matrix with " }{TEXT 1132 6 "moduli" }{TEXT -1 22 " of the elements of [" }{TEXT 1133 1 "Z" }{TEXT -1 24 "] round ed-off to, say, " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 21 " signif icant digits:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "`abs(Z)` : = evalf(matrix(`abs(Z)`), 4) : abs(Z) = matrix(`abs(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"ZG-%'matrixG6#7%7%$\"%*4&!\"$$ \"%CjF/$\"%;wF/7%$\"%W*)F/$\"%I5!\"#$\"%m6F97%$\"%/8F9$\"%U9F9$\"%\"e \"F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 75 "Note that the same effect may be obtained setting the e nvironment variable " }{TEXT 575 6 "Digits" }{TEXT -1 5 " to " } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 42 ", i.e. using the command li ne as follows:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 576 85 "> Digits := 4 \+ : `abs(Z)` := evalf(matrix(`abs(Z)`)) : abs(Z) = matrix(`abs(Z)`) \+ ;" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1134 6 "Step 2" }{TEXT -1 24 ". Obtain a matrix with " }{TEXT 573 20 "principal arguments " }{TEXT -1 1 " " }{TEXT 1136 1 "(" } {TEXT -1 4 "in " }{TEXT 1138 7 "degrees" }{TEXT 1137 1 ")" }{TEXT -1 22 " of the elements of [" }{TEXT 1135 1 "Z" }{TEXT -1 24 "] rounded- off to, say, " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 21 " significa nt digits:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "`deg(arg(Z))` := map(x->evalf(x, 3), `deg(arg(Z))`) : deg(Arg(Z)) = matrix(`deg(a rg(Z))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$degG6#-%$ArgG6#%\"ZG -%'matrixG6#7%7%$\"$(y!\"\"$\"$;(F2$!$o'F27%$\"$M'F2$!$4'F2$\"$!fF27%$ \"$v&F2$\"$j&F2$!$`&F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1139 6 "Step 3" }{TEXT -1 65 ". Define an auxiliary matrix of the same size as that of matrix [" }{TEXT 1140 1 "Z" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "z_p_ e := matrix(rowdim(Z), coldim(Z)) :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1142 6 "Step 4" }{TEXT -1 22 " . Construct a matrix [" }{TEXT 1144 7 "Z_p_eng" }{TEXT -1 59 "] contai ning \"engineering\" polar forms of the elements of [" }{TEXT 1141 1 " Z" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This may be done using any of the followi ng methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1143 8 "Method 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "z_p_e1 := map(x->x * `@`, `abs(Z)`) : z_p_e2 := map (x->[x * `\260`], `deg(arg(Z))`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "for i to rowdim(Z) do for j to coldim(Z) do z_p_e[i ,j] := z_p_e1[i,j]*z_p_e2[i,j] : od : od :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "Z_p_eng := matrix(z_p_e) : Z_p_eng = matrix( Z_p_eng) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(Z_p_engG-%'matrixG6#7 %7%,$*&%\"@G\"\"\"7#,$%\"|[vG$\"$(y!\"\"F-$\"%*4&!\"$,$*&F,F-7#,$F0$\" $;(F3F-$\"%CjF6,$*&F,F-7#,$F0$!$o'F3F-$\"%;wF67%,$*&F,F-7#,$F0$\"$M'F3 F-$\"%W*)F6,$*&F,F-7#,$F0$!$4'F3F-$\"%I5!\"#,$*&F,F-7#,$F0$\"$!fF3F-$ \"%m6FX7%,$*&F,F-7#,$F0$\"$v&F3F-$\"%/8FX,$*&F,F-7#,$F0$\"$j&F3F-$\"%U 9FX,$*&F,F-7#,$F0$!$`&F3F-$\"%\"e\"FX" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Any element of th is matrix may be easily extracted and so may be its numerical componen ts, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "z_p_eng[32] := \+ Z_p_eng[3,2] : 'z_p_eng[32]' = z_p_eng[32] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%(z_p_engG6#\"#K,$*&%\"@G\"\"\"7#,$%\"|[vG$\"$j&!\"\" F+$\"%U9!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`abs(z_p_eng)`[32] := op(1, z_p_eng[32]) \+ : Abs('z_p_eng[32]') = `abs(z_p_eng)`[32] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AbsG6#&%(z_p_engG6#\"#K$\"%U9!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "` deg(arg(z_p_eng))`[32] := op(op(3, z_p_eng[32]))/`\260` :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "deg(Arg('z_p_eng[32]')) = `deg(arg( z_p_eng))`[32] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$degG6#-%$ArgG6 #&%(z_p_engG6#\"#K$\"$j&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1145 8 "Method 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "z_p_e1 := map(x->x * `@`, `abs(Z)`) : z_p _e2 := map(x->angle(x * `\260`), `deg(arg(Z))`) :" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 99 "for i to rowdim(Z) do for j to coldim(Z) do z_p_e[i,j] := z_p_e1[i,j]*z_p_e2[i,j] : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Z_p_eng := matrix(z_p_e) : Z_p_en g = matrix(Z_p_eng) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(Z_p_engG-% 'matrixG6#7%7%,$*&%\"@G\"\"\"-%&angleG6#,$%\"|[vG$\"$(y!\"\"F-$\"%*4&! \"$,$*&F,F--F/6#,$F2$\"$;(F5F-$\"%CjF8,$*&F,F--F/6#,$F2$!$o'F5F-$\"%;w F87%,$*&F,F--F/6#,$F2$\"$M'F5F-$\"%W*)F8,$*&F,F--F/6#,$F2$!$4'F5F-$\"% I5!\"#,$*&F,F--F/6#,$F2$\"$!fF5F-$\"%m6Fhn7%,$*&F,F--F/6#,$F2$\"$v&F5F -$\"%/8Fhn,$*&F,F--F/6#,$F2$\"$j&F5F-$\"%U9Fhn,$*&F,F--F/6#,$F2$!$`&F5 F-$\"%\"e\"Fhn" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Any element of this matrix may be easily \+ extracted and so may be its numerical components, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "z_p_eng[13] := Z_p_eng[1,3] : 'z_ p_eng[13]' = z_p_eng[13] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%(z_p_ engG6#\"#8,$*&%\"@G\"\"\"-%&angleG6#,$%\"|[vG$!$o'!\"\"F+$\"%;w!\"$" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`abs(z_p_eng)`[13] := op(1, z_p_eng[13]) : Abs('z_p _eng[13]') = `abs(z_p_eng)`[13] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$AbsG6#&%(z_p_engG6#\"#8$\"%;w!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`deg(arg(z_p_eng) )`[13] := op(op(3, z_p_eng[13]))/`\260` :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "deg(Arg('z_p_eng[13]')) = `deg(arg(z_p_eng))`[13] ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$degG6#-%$ArgG6#&%(z_p_engG6#\" #8$!$o'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1146 8 "Method 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "z_p_e1 := map(x->x * `|`, `abs(Z)`) : z_p_e2 := map (x->[x * `\260`], `deg(arg(Z))`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "for i to rowdim(Z) do for j to coldim(Z) do z_p_e[i ,j] := z_p_e1[i,j]*z_p_e2[i,j] : od : od :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "Z_p_eng := matrix(z_p_e) : Z_p_eng = matrix( Z_p_eng) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(Z_p_engG-%'matrixG6#7 %7%,$*&%\"|grG\"\"\"7#,$%\"|[vG$\"$(y!\"\"F-$\"%*4&!\"$,$*&F,F-7#,$F0$ \"$;(F3F-$\"%CjF6,$*&F,F-7#,$F0$!$o'F3F-$\"%;wF67%,$*&F,F-7#,$F0$\"$M' F3F-$\"%W*)F6,$*&F,F-7#,$F0$!$4'F3F-$\"%I5!\"#,$*&F,F-7#,$F0$\"$!fF3F- $\"%m6FX7%,$*&F,F-7#,$F0$\"$v&F3F-$\"%/8FX,$*&F,F-7#,$F0$\"$j&F3F-$\"% U9FX,$*&F,F-7#,$F0$!$`&F3F-$\"%\"e\"FX" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Any element of th is matrix may be easily extracted and so may be its numerical componen ts, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "z_p_eng[11] := \+ Z_p_eng[1,1] : 'z_p_eng[11]' = z_p_eng[11] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%(z_p_engG6#\"#6,$*&%\"|grG\"\"\"7#,$%\"|[vG$\"$(y!\" \"F+$\"%*4&!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`abs(z_p_eng)`[11] := op(1, z_p_eng [11]) : Abs('z_p_eng[11]') = `abs(z_p_eng)`[11] ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$AbsG6#&%(z_p_engG6#\"#6$\"%*4&!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`deg(arg(z_p_eng))`[11] := op(op(3, z_p_eng[11]))/`\260` :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "deg(Arg('z_p_eng[11]')) = `d eg(arg(z_p_eng))`[11] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$degG6#- %$ArgG6#&%(z_p_engG6#\"#6$\"$(y!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Each method serves the \+ purpose equally well." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 258 "" 0 "" {TEXT 574 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 393 4 "N.B." }{TEXT -1 56 " If the elements of a complex matrix are given in the " } {TEXT 398 5 "polar" }{TEXT -1 19 " form containing " }{TEXT 399 15 " decimal numbers" }{TEXT -1 77 " and it is required to display them in this form, the list of the elements " }{TEXT 19 4 "must" }{TEXT -1 67 " be enclosed in single quotes to prevent internal evaluation, e.g ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "A := matrix(2, 2, '[4 *exp(0.4*I), -0.7*exp(-0.6*I), -4*exp(0.5*I), 2.3*exp(0.2*I)]') : A \+ = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7 $,$-%$expG6#^#$\"\"%!\"\"F0,$-F,6#^#$!\"'F1$!\"(F17$,$-F,6#^#$\"\"&F1! \"%,$-F,6#^#$\"\"#F1$\"#BF1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "otherwise " }{TEXT 940 5 "Maple " }{TEXT -1 69 " would evaluate all the elements internally and displa y them in the " }{TEXT 939 14 "canonical form" }{TEXT -1 26 " (recta ngular form), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A = m atrix(2, 2, [4*exp(0.4*I), -0.7*exp(-0.6*I), -4*exp(0.5*I), 2.3*exp(0. 2*I)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$^$$ \"+wRC%o$!\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5NF-$!+a@ q<>F-^$$\"+HJ:aAF-$\"+3YRpXF3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Matrix [" }{TEXT 952 1 "A" } {TEXT -1 112 "] defined and input with the element list enclosed in si ngle quotes may then be evaluated and displayed in the " }{TEXT 978 14 "canonical form" }{TEXT -1 49 " using any of the following alterna tive methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 936 8 "Method 1" }{TEXT -1 21 ". Using the funct ion " }{TEXT 933 5 "evalf" }{TEXT -1 26 " and any of the functions " } {TEXT 979 5 "evalm" }{TEXT -1 2 ", " }{TEXT 980 6 "matrix" }{TEXT -1 5 ", or " }{TEXT 981 2 "op" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "A = evalf(evalm(A)) ; A = evalf(matrix(A)) ; A \+ = evalf(op(A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6# 7$7$^$$\"+wRC%o$!\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5N F-$!+a@q<>F-^$$\"+HJ:aAF-$\"+3YRpXF3" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"AG-%'matrixG6#7$7$^$$\"+wRC%o$!\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+ 9t\\_RF37$^$$!+[-L5NF-$!+a@q<>F-^$$\"+HJ:aAF-$\"+3YRpXF3" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$^$$\"+wRC%o$!\"*$\"+pLnd:F -^$$!+/$\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5NF-$!+a@q<>F-^$$\"+HJ:aAF-$\"+3 YRpXF3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 937 8 "Method 2" }{TEXT -1 21 ". Using the function " } {TEXT 930 7 "convert" }{TEXT -1 31 " together with the form (type) " } {TEXT 931 5 "float" }{TEXT -1 2 " (" }{TEXT 932 5 "float" }{TEXT -1 43 "ing-point number) and any of the functions " }{TEXT 982 5 "evalm" }{TEXT -1 2 ", " }{TEXT 983 6 "matrix" }{TEXT -1 5 ", or " }{TEXT 984 2 "op" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "A \+ = convert(evalm(A), float) ; A = convert(matrix(A), float) ; A = c onvert(op(A), float) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'mat rixG6#7$7$^$$\"+wRC%o$!\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+9t\\_RF37$^$$! +[-L5NF-$!+a@q<>F-^$$\"+HJ:aAF-$\"+3YRpXF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$^$$\"+wRC%o$!\"*$\"+pLnd:F-^$$!+/ $\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5NF-$!+a@q<>F-^$$\"+HJ:aAF-$\"+3YRpXF3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$^$$\"+wRC%o$ !\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5NF-$!+a@q<>F-^$$ \"+HJ:aAF-$\"+3YRpXF3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 938 8 "Method 3" }{TEXT -1 21 ". Using th e function " }{TEXT 934 3 "map" }{TEXT -1 71 " together with the arrow -type procedure including any of the functions " }{TEXT 935 4 "eval" } {TEXT -1 2 ", " }{TEXT 975 5 "evalc" }{TEXT -1 2 ", " }{TEXT 976 5 "ev alf" }{TEXT -1 5 ", or " }{TEXT 977 5 "value" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "A = map(x->eval(x), A) ; \+ A = map(x->evalc(x), A) ; A = map(x->evalf(x), A) ; A = map(x->val ue(x), A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$^ $$\"+wRC%o$!\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5NF-$!+ a@q<>F-^$$\"+HJ:aAF-$\"+3YRpXF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"AG-%'matrixG6#7$7$^$$\"+wRC%o$!\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+9t\\ _RF37$^$$!+[-L5NF-$!+a@q<>F-^$$\"+HJ:aAF-$\"+3YRpXF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$^$$\"+wRC%o$!\"*$\"+pLnd:F-^$$ !+/$\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5NF-$!+a@q<>F-^$$\"+HJ:aAF-$\"+3YRpX F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$^$$\"+wRC% o$!\"*$\"+pLnd:F-^$$!+/$\\tx&!#5$\"+9t\\_RF37$^$$!+[-L5NF-$!+a@q<>F-^$ $\"+HJ:aAF-$\"+3YRpXF3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "where the function " }{TEXT 970 5 "evalc" }{TEXT -1 28 " is specifically designed to" }{TEXT 973 1 " " } {TEXT 971 4 "eval" }{TEXT -1 5 "uate " }{TEXT 972 1 "c" }{TEXT -1 13 " omplex-valued" }{TEXT 974 1 " " }{TEXT -1 12 "expressions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "T he same applies to matrix elements given in any other " }{TEXT 397 14 "transcendental" }{TEXT -1 19 " form containing " }{TEXT 400 15 " decimal numbers" }{TEXT -1 7 ", e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "B := matrix(2, 2, '[2*sin(0.2-0.3*I), 0.4*cos(0.4+0. 6*I), 1.1*cos(0.5-0.7*I), -5*sin(0.6+0.2*I)]') :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$,$-%$sinG6#^$$\"\"#!\"\"$!\"$F1F0,$-%$cosG 6#^$$\"\"%F1$\"\"'F1F97$,$-F66#^$$\"\"&F1$!\"(F1$\"#6F1,$-F,6#^$F;F/! \"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 124 "since not enclosing the element list in single quotes \+ would display the matrix with the elements in the canonical form, i.e. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "B = matrix(2, 2, [2*sin (0.2-0.3*I), 0.4*cos(0.4+0.6*I), 1.1*cos(0.5-0.7*I), -5*sin(0.6+0.2*I) ]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$^$$\"+iS ``T!#5$!+QK+pfF-^$$\"+)3VvO%F-$!+/L)p\"**!#67$^$$\"+vem67!\"*$\"+)RG0+ %F-^$$!+!3l)zGF:$!+v')[3$)F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Subsequent floating-point evalu ation of matrix [" }{TEXT 951 1 "B" }{TEXT -1 74 "] defined and input \+ with the element list enclosed in single quotes yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B = evalf(matrix(B)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$^$$\"+iS``T!#5$!+QK+pfF-^ $$\"+)3VvO%F-$!+/L)p\"**!#67$^$$\"+vem67!\"*$\"+)RG0+%F-^$$!+!3l)zGF:$ !+v')[3$)F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 394 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 395 4 "N.B." }{TEXT -1 56 " If the \+ elements of a complex matrix are given in the " }{TEXT 465 5 "polar" }{TEXT -1 19 " form containing " }{TEXT 396 10 "irrational" }{TEXT -1 85 " numbers and it is required to display them in this form, the \+ list of the elements " }{TEXT 19 4 "must" }{TEXT -1 67 " be enclosed in single quotes to prevent internal evaluation, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "C := matrix(2, 2, '[3*exp(Pi/3*I), -3/5*exp(Pi/4*I), 4/7*exp(Pi/5*I), 5*exp(Pi/6*I)]') : C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7$7$,$-%$expG6 #*&^##\"\"\"\"\"$F1%#PiGF1F2,$-F,6#*&^##F1\"\"%F1F3F1#!\"$\"\"&7$,$-F, 6#*&^##F1F=F1F3F1#F:\"\"(,$-F,6#*&^##F1\"\"'F1F3F1F=" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "otherwi se " }{TEXT 941 5 "Maple" }{TEXT -1 60 " would evaluate the elements i nternally and display them as " }{TEXT 942 5 "exact" }{TEXT -1 14 " nu mbers, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "C = matrix(2 , 2, [3*exp(Pi/3*I), -3/5*exp(Pi/4*I), 4/7*exp(Pi/5*I), 5*exp(Pi/6*I)] ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7$7$,&#\"\"$ \"\"#\"\"\"*&^#F+F.-%%sqrtG6#F,F.F.,&*$-F26#F-F.#!\"$\"#5*&^#F8F.F6F.F .7$,$-%$expG6#*&^##F.\"\"&F.%#PiGF.#\"\"%\"\"(,&*$F1F.#FEF-^#FLF." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "A further " }{TEXT 945 5 "exact" }{TEXT -1 49 " symbolic evalua tion of the element at location " }{TEXT 946 5 "(2,1)" }{TEXT -1 57 " is possible, which may be performed using the function " }{TEXT 943 3 "map" }{TEXT -1 59 " together with the arrow-type procedure includin g function " }{TEXT 944 5 "evalc" }{TEXT -1 6 ", i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "C := map(x->evalc(x), C) : C = ma trix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7$7$,&# \"\"$\"\"#\"\"\"*&^#F+F.-%%sqrtG6#F,F.F.,&*$-F26#F-F.#!\"$\"#5*&^#F8F. F6F.F.7$,&-%$cosG6#,$%#PiG#F.\"\"&#\"\"%\"\"(*&^#FFF.-%$sinGFAF.F.,&*$ F1F.#FEF-^#FOF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Floating-point evaluation of the matrix y ields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "C = evalf(matrix(C )) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7$7$^$$\"+++ ++:!\"*$\"+7i2)f#F-^$$!+'oSEC%!#5F17$^$$\"+`U&Hi%F3$\"+rGxeLF3^$$\"+?q 7IVF-$\"+++++DF-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 950 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 947 4 "N.B." }{TEXT -1 62 " I f the elements of a complex matrix are given in any other " }{TEXT 948 14 "transcendental" }{TEXT -1 19 " form containing " }{TEXT 949 10 "irrational" }{TEXT -1 90 " numbers, and it is required to display them in this form, the list of the elements need " }{TEXT 953 3 "not " }{TEXT -1 35 " be enclosed in single quotes, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "E := matrix(2, 2, [6*sin(1/5-Pi/4*I), -s qrt(3)*cos(2/5+Pi/3*I), 2*cos(1/2-Pi/5*I), -5*sin(3/7+Pi/6*I)]) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "E = matrix(E) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$7$,$-%$sinG6#,&#\"\"\"\"\"& F0*&^##!\"\"\"\"%F0%#PiGF0F0\"\"',$*&-%%sqrtG6#\"\"$F0-%$cosG6#,&#\"\" #F1F0*&^##F0F>F0F7F0F0F0F57$,$-F@6#,&#F0FDF0*&^##F5F1F0F7F0F0FD,$-F,6# ,&#F>\"\"(F0*&^##F0F8F0F7F0F0!\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "since the " }{TEXT 955 6 " matrix" }{TEXT -1 15 " function will " }{TEXT 954 3 "not" }{TEXT -1 21 " perform any further " }{TEXT 956 5 "exact" }{TEXT -1 89 " evaluat ion internally and the matrix will be displayed in the unaltered exact form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "E = matrix( 2, 2, [6*sin(1/5-Pi/4*I), -sqrt(3)*cos(2/5+Pi/3*I), 2*cos(1/2-Pi/5*I), -5*sin(3/7+Pi/6*I)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'ma trixG6#7$7$,$-%$sinG6#,&#\"\"\"\"\"&F0*&^##!\"\"\"\"%F0%#PiGF0F0\"\"', $*&-%%sqrtG6#\"\"$F0-%$cosG6#,&#\"\"#F1F0*&^##F0F>F0F7F0F0F0F57$,$-F@6 #,&#F0FDF0*&^##F5F1F0F7F0F0FD,$-F,6#,&#F>\"\"(F0*&^##F0F8F0F7F0F0!\"& " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "A further " }{TEXT 959 5 "exact" }{TEXT -1 96 " symbolic \+ evaluation of the matrix elements is possible and may be performed usi ng the function " }{TEXT 957 3 "map" }{TEXT -1 59 " together with the \+ arrow-type procedure including function " }{TEXT 958 5 "evalc" }{TEXT -1 6 ", i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "E := map(x- >evalc(x), E) : E = matrix(E) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"EG-%'matrixG6#7$7$,&*&-%$sinG6##\"\"\"\"\"&F0-%%coshG6#,$%#PiG#F0\" \"%F0\"\"'*(^#!\"'F0-%$cosGF.F0-%%sinhGF4F0F0,&*(-%%sqrtG6#\"\"$F0-F>6 ##\"\"#F1F0-F36#,$F6#F0FFF0!\"\"**^#F0F0FCF0-F-FHF0-F@FLF0F07$,&*&-F>6 ##F0FJF0-F36#,$F6F/F0FJ*(^#FJF0-F-FXF0-F@FenF0F0,&*&-F-6##FF\"\"(F0-F3 6#,$F6#F0F9F0!\"&*(^#FeoF0-F>F^oF0-F@FboF0F0" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Floating-point \+ evaluation of the matrix yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "E = evalf(matrix(E)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" EG-%'matrixG6#7$7$^$$\"+3_&*y:!\"*$!+gA83^F-^$$!+Kn(Hb#F-$\"+J_)oU)!#5 7$^$$\"+?)pJ6#F-$\"+SI%*GkF5^$$!+rZDpBF-$!+$**G:\\#F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 401 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 926 4 "N.B." }{TEXT -1 60 " The following matrix formed fr om elements containing the " }{TEXT 960 14 "imaginary unit" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "F := matrix(2, 3, '[I^(I/2), I^I, I^(2*I), exp(I*Pi), exp(2*I*Pi), cosh(I)]') : F = matrix(F) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"FG-%'matrixG6#7$7%)^#\"\"\"^## F,\"\"#)F+F+)F+^#F/7%-%$expG6#*&F+F,%#PiGF,-F56#*&F2F,F8F,-%%coshG6#F+ " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "evaluates to a " }{TEXT 927 11 "real matrix" }{TEXT -1 32 " as evidenced by the following " }{TEXT 1101 5 "exact" }{TEXT -1 8 " result," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F := map(x-> evalc(x), F) : F = matrix(F) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"FG-%'matrixG6#7$7%-%$expG6#,$%#PiG#!\"\"\"\"%-F+6#,$F.#F0\"\"#-F+6#, $F.F07%F0\"\"\"-%$cosG6#F;" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "or by its the floating-point appr oximation," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F := evalf(ma trix(F)) : F = matrix(F) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"FG -%'matrixG6#7$7%$\"+x7QfX!#5$\"+jdzy?F,$\"+D=R@V!#67%$!\"\"\"\"!$\"\" \"F5$\"+fI-.aF," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 928 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 929 4 "N.B." }{TEXT -1 60 " T he following matrix formed from elements containing the " }{TEXT 961 14 "imaginary unit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "G := \+ matrix(2, 2, '[exp(I*Pi/2), (I/2)^I, exp(I), sinh(I)]') : G = matri x(G) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"GG-%'matrixG6#7$7$-%$exp G6#*&^##\"\"\"\"\"#F0%#PiGF0)F.^#F07$-F+6#F4-%%sinhGF7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "e valuates to a " }{TEXT 1102 14 "complex matrix" }{TEXT -1 32 " as ev idenced by the following " }{TEXT 1103 5 "exact" }{TEXT -1 8 " result, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "G := map(x->evalc(x), G ) : G = matrix(G) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"GG-%'matr ixG6#7$7$^#\"\"\",&*&-%$expG6#,$%#PiG#!\"\"\"\"#F+-%$cosG6#-%#lnG6#F5F +F+*(^#F4F+F.F+-%$sinGF8F+F+7$,&-F76#F+F+*&F*F+-F?FCF+F+FD" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "o r by its floating-point approximation," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "G := evalf(matrix(G)) : G = matrix(G) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"GG-%'matrixG6#7$7$^#$\"\"\"\"\"!^$$\"+p0 4*f\"!#5$!+%**p#G8F17$^$$\"+fI-.aF1$\"+[)4ZT)F1^#F8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 1008 5 "* * \+ *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1015 4 "N.B." }{TEXT -1 5 " A " }{TEXT 1019 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 1018 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$ " }{TEXT 1020 1 ")" }{TEXT -1 46 " complex matrix, which satisfies th e equation" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 1012 1 " k" }{TEXT -1 1 "]" }{TEXT 1009 4 "^(3 " }{TEXT 1010 1 "n" }{TEXT 1011 1 ")" }{TEXT -1 1 " " }{TEXT 1013 1 "=" }{TEXT -1 2 " [" }{TEXT 1014 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where " }{TEXT 1017 1 "n" }{TEXT -1 56 " is any (positive or negative) integer, is called the " } {TEXT 1016 8 "k-matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Let a " }{TEXT 1022 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 1021 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 1023 1 ")" }{TEXT -1 28 " complex \+ matrix be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k := \+ matrix(3,3, [0, 0, -I, I, 0, 0, 0, 1, 0]) : k = matrix(k) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"kG-%'matrixG6#7%7%\"\"!F*^#!\"\"7% ^#\"\"\"F*F*7%F*F/F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Exemplarily, let " }{XPPEDIT 18 0 "n=1" "6#/%\"nG\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "n=-2" " 6#/%\"nG,$\"\"#!\"\"" }{TEXT -1 13 ". Compute [" }{TEXT 1027 1 "k" } {TEXT -1 1 "]" }{TEXT 1024 4 "^(3 " }{TEXT 1025 1 "n" }{TEXT 1026 1 ") " }{TEXT -1 22 " for both values of " }{TEXT 1028 1 "n" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "n := 1 : `k^3*1` := evalm(k^(3*n)) : n := -2 : `k^3*(-2)` := evalm(k^(3*n)) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "k^3 = matrix(`k^3*1`) ; k^ ` -6` = matrix(`k^3*(-2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)% \"kG\"\"$\"\"\"-%'matrixG6#7%7%F(\"\"!F.7%F.F(F.7%F.F.F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/)%\"kG%$~-6G-%'matrixG6#7%7%\"\"\"\"\"!F-7%F-F, F-7%F-F-F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Notice that the " }{TEXT 1033 8 "k-matrix" } {TEXT -1 31 " may be considered to be the " }{TEXT 1045 9 "cube root " }{TEXT -1 8 " of a " }{TEXT 1042 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$ " }{TEXT 1041 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 1043 1 ") " }{TEXT -1 2 " " }{TEXT 1044 4 "unit" }{TEXT -1 10 " matrix [" } {TEXT 1046 1 "U" }{TEXT -1 5 "], or" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 1070 1 "k" }{TEXT -1 1 "]" }{TEXT 1071 3 " = " } {TEXT -1 1 "[" }{TEXT 1072 1 "U" }{TEXT -1 1 "]" }{TEXT 1073 6 "^(1/3) " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "By definition, the cube root of a matrix [" }{TEXT 1069 1 "U" }{TEXT -1 16 "] is a matrix [" }{TEXT 1068 1 "U" }{TEXT -1 1 "] " }{TEXT 1067 6 "^(1/3)" }{TEXT -1 26 " having the property that" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 1034 1 "U" }{TEXT -1 1 "]" }{TEXT 1039 4 " = \{" }{TEXT -1 1 "[" }{TEXT 1035 1 "U" }{TEXT -1 1 "]" }{TEXT 1038 7 "^(1/3)\}" }{TEXT -1 1 " " }{TEXT 1040 1 "\{" } {TEXT -1 1 "[" }{TEXT 1036 1 "U" }{TEXT -1 1 "]" }{TEXT 1037 7 "^(1/3) \}" }{TEXT -1 1 " " }{TEXT 1049 1 "\{" }{TEXT -1 1 "[" }{TEXT 1047 1 " U" }{TEXT -1 1 "]" }{TEXT 1048 7 "^(1/3)\}" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Setting" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 1050 1 "U" }{TEXT -1 1 "]" }{TEXT 1051 9 "^(1/3) = " }{TEXT -1 1 "[" }{TEXT 1052 1 "k" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "and substituting it into the above yields " }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 1053 1 "U" }{TEXT -1 1 "]" }{TEXT 1054 3 " = " }{TEXT -1 1 "[" }{TEXT 1055 1 "k" }{TEXT -1 3 "] [" }{TEXT 1056 1 "k" }{TEXT -1 3 "] [" }{TEXT 1057 1 "k" } {TEXT -1 2 "] " }{TEXT 1058 1 "=" }{TEXT -1 2 " [" }{TEXT 1059 1 "k" } {TEXT -1 1 "]" }{TEXT 1060 2 "^3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 1061 8 "k-mat rix" }{TEXT -1 5 " is " }{TEXT 1062 3 "not" }{TEXT -1 4 " a " } {TEXT 1063 6 "unique" }{TEXT -1 16 " cube root of [" }{TEXT 1064 1 "U " }{TEXT -1 25 "] since the unit matrix [" }{TEXT 1065 1 "U" }{TEXT -1 33 "] is itself also a cube root of [" }{TEXT 1066 1 "U" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Notice that the determinant of the " }{TEXT 1029 8 "k -matrix" }{TEXT -1 6 " is " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 7 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(k)` := det(k ) : Det(k) = `det(k)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6 #%\"kG\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 1030 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1076 4 "N.B." }{TEXT -1 24 " The m atrix product of " }{TEXT 1166 7 "complex" }{TEXT -1 16 " matrices wit h " }{TEXT 1195 8 "symbolic" }{TEXT -1 63 " elements may be presente d as the sum of a matrix comprising " }{TEXT 1074 4 "real" }{TEXT -1 33 " parts and a matrix comprising " }{TEXT 1075 9 "imaginary" } {TEXT -1 42 " parts of elements of the product matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "T he method presented hereunder is " }{TEXT 1164 9 "universal" }{TEXT -1 43 ", i.e. it works for complex matrices with " }{TEXT 1165 8 "sym bolic" }{TEXT -1 101 " elements, which satisfy the multiplication con formability rule. The matrix elements may have both " }{TEXT 1167 4 " real" }{TEXT -1 7 " and " }{TEXT 1168 9 "imaginary" }{TEXT -1 38 " \+ parts, or some of them may be only " }{TEXT 1169 4 "real" }{TEXT -1 6 " or " }{TEXT 1170 14 "pure imaginary" }{TEXT -1 7 ", or " } {XPPMATH 20 "6#%&zerosG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "For simplicity in display, consider square " }{TEXT 1171 1 "(" }{XPPEDIT 18 0 "2" "6# \"\"#" }{TEXT 1172 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1173 1 ")" }{TEXT -1 20 " complex matrices [" }{TEXT 1174 2 "Z1" } {TEXT -1 7 "] and [" }{TEXT 1175 2 "Z2" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Z1 := matrix(2, 2, [a[11]+b[ 11]*I, b[12]*I, 0, a[22]+b[22]*I]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Z2 := matrix(2, 2, [c[11], c[12]+d[12]*I, d[21]*I, 0] ) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Z1 = matrix(Z1) ; \+ Z2 = matrix(Z2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#Z1G-%'matrixG6 #7$7$,&&%\"aG6#\"#6\"\"\"*&^#F/F/&%\"bGF-F/F/*&F1F/&F36#\"#7F/7$\"\"!, &&F,6#\"#AF/*&F1F/&F3F " 0 "" {MPLTEXT 1 0 69 "`Z1 Z2` := map(expand , evalm(Z1 &* Z2)) : Z1*Z2 = matrix(`Z1 Z2`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#Z1G\"\"\"%#Z2GF&-%'matrixG6#7$7$,(*&&%\"cG6#\"#6F& &%\"aGF1F&F&*(^#F&F&F/F&&%\"bGF1F&F&*&&F86#\"#7F&&%\"dG6#\"#@F&!\"\",* *&F3F&&F0F;F&F&*(F6F&F3F&&F>F;F&F&*(F6F&F7F&FDF&F&*&F7F&FFF&FA7$,&*(F6 F&F=F&&F46#\"#AF&F&*&F=F&&F8FMF&FA\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [" } {TEXT 1098 2 "Z1" }{TEXT -1 3 "] [" }{TEXT 1099 2 "Z2" }{TEXT -1 29 "] may be presented as either" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Z1*Z2 = Real(Z1*Z2) + I*Imaginary(Z1*Z2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#Z1G\"\"\"%#Z2GF&,&-%%RealG6#F$F&*&^#F&F&-%*Imagina ryGF+F&F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 23 "or, in the matrix form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "R_e := matrix(rowdim(`Z1 Z2`), coldim(`Z1 Z2`)) \+ : I_m := matrix(rowdim(`Z1 Z2`), coldim(`Z1 Z2`)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "for i to rowdim(`Z1 Z2`) do for j to co ldim(`Z1 Z2`) do R_e[i,j] := re[i,j] : I_m[i,j] := im[i,j] : od \+ : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "R_e := matrix(R_ e) : I_m := matrix(I_m) : Z1*Z2 = matrix(R_e) + I*matrix(I_m) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#Z1G\"\"\"%#Z2GF&,&-%'matrixG6#7$ 7$&%#reG6$F&F&&F/6$F&\"\"#7$&F/6$F3F&&F/6$F3F3F&*&^#F&F&-F*6#7$7$&%#im GF0&F@F27$&F@F6&F@F8F&F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "To determine the real elements " }{XPPEDIT 18 0 "re[i,j]" "6#&%#reG6$%\"iG%\"jG" }{TEXT -1 27 " and i maginary elements " }{XPPEDIT 18 0 "im[i,j]" "6#&%#imG6$%\"iG%\"jG" } {TEXT -1 52 " of the component matrices, do the following steps." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1085 6 "Step 1" }{TEXT -1 40 ". Convert each element of the product [ " }{TEXT 1083 2 "Z1" }{TEXT -1 3 "] [" }{TEXT 1084 2 "Z2" }{TEXT -1 60 "] to a list comprising individual terms of a given element:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "for i to rowdim(`Z1 Z2`) do for j to coldim(`Z1 Z2`) do L[i,j] := convert(`Z1 Z2`[i,j], list) \+ : print(evaln(L[i,j]) = L[i,j]) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6$\"\"\"F'7%*&&%\"cG6#\"#6F'&%\"aGF,F'*(^#F'F'F* F'&%\"bGF,F',$*&&F36#\"#7F'&%\"dG6#\"#@F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6$\"\"\"\"\"#7&*&&%\"aG6#\"#6F'&%\"cG6#\"#7F'*(^ #F'F'F+F'&%\"dGF1F'*(F4F'&%\"bGF-F'F/F',$*&F8F'F5F'!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"LG6$\"\"#\"\"\"7$*(^#F(F(&%\"dG6#\"#@F(&%\" aG6#\"#AF(,$*&F,F(&%\"bGF2F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"LG6$\"\"#F'7#\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1086 6 "Step 2" }{TEXT -1 66 ". Rename th e lists by indexing each with a single-digit subscript:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "n := 0 : for i to rowdim(`Z1 Z2` ) do for j to coldim(`Z1 Z2`) do n := n+1 : L[n] := L[i,j] : pri nt(evaln(L[n]) = L[i,j]) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"\"7%*&&%\"cG6#\"#6F'&%\"aGF,F'*(^#F'F'F*F'&%\"bGF,F ',$*&&F36#\"#7F'&%\"dG6#\"#@F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"LG6#\"\"#7&*&&%\"aG6#\"#6\"\"\"&%\"cG6#\"#7F.*(^#F.F.F*F.&%\"dGF 1F.*(F4F.&%\"bGF,F.F/F.,$*&F8F.F5F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"$7$*(^#\"\"\"F+&%\"dG6#\"#@F+&%\"aG6#\"#AF+,$*&F,F+ &%\"bGF2F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"%7#\" \"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 1087 6 "Step 3" }{TEXT -1 30 ". Assign a temporary initial " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 69 " value to each real and imagin ary element of the component matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "for i to rowdim(`Z1 Z2`) do for j to coldim(`Z1 Z2` ) do re[i,j] := 0 : im[i,j] := 0 : print(evaln(re[i,j]) = re[i,j] , ` ` * evaln(im[i,j]) = im[i,j]) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%#reG6$\"\"\"F'\"\"!/*&%%~~~~GF'&%#imGF&F'F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%#reG6$\"\"\"\"\"#\"\"!/*&%%~~~~GF'& %#imGF&F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%#reG6$\"\"#\"\"\"\" \"!/*&%%~~~~GF(&%#imGF&F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%#reG 6$\"\"#F'\"\"!/*&%%~~~~G\"\"\"&%#imGF&F,F(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1088 6 "Step 4" }{TEXT -1 67 ". Isolate real and imaginary elements from each of the above li sts:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "n := 0 : for i t o rowdim(`Z1 Z2`) do for j to coldim(`Z1 Z2`) do n := n+1 : for m \+ to nops(L[n]) do if L[n][m] = 0 then re[i,j] := 0 : im[i,j] := 0 \+ elif sign((L[n][m])^2/abs(L[n][m])) = 1 then re[i,j] := re[i,j] + L[ n][m] else im[i,j] := im[i,j] + L[n][m] fi : if m = nops(L[n]) the n print(evaln(re[i,j]) = re[i,j]) : print(evaln(im[i,j]) = im[i,j]) fi : od : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#reG 6$\"\"\"F',&*&&%\"cG6#\"#6F'&%\"aGF,F'F'*&&%\"bG6#\"#7F'&%\"dG6#\"#@F' !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#imG6$\"\"\"F'*(^#F'F'&%\" cG6#\"#6F'&%\"bGF,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#reG6$\"\" \"\"\"#,&*&&%\"aG6#\"#6F'&%\"cG6#\"#7F'F'*&&%\"bGF-F'&%\"dGF1F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#imG6$\"\"\"\"\"#,&*(^#F'F'&%\"aG 6#\"#6F'&%\"dG6#\"#7F'F'*(F+F'&%\"bGF.F'&%\"cGF2F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#reG6$\"\"#\"\"\",$*&&%\"dG6#\"#@F(&%\"bG6#\"#AF( !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#imG6$\"\"#\"\"\"*(^#F(F(& %\"dG6#\"#@F(&%\"aG6#\"#AF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#reG 6$\"\"#F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#imG6$\"\"#F'\"\" !" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1089 6 "Step 5" }{TEXT -1 15 ". Compute the " }{TEXT 1091 5 "va lue" }{TEXT -1 53 " of each element of the real and imaginary matrice s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "for i to rowdim(`Z1 \+ Z2`) do for j to coldim(`Z1 Z2`) do R_e[i,j] := re[i,j] : re[i,j] \+ := R_e[i,j] : I_m[i,j] := evalc(-I*im[i,j]) : im[i,j] := I_m[i,j] \+ : od : od :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1092 1 "\225" }{TEXT -1 40 " values of elements of the real matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "fo r i to rowdim(`Z1 Z2`) do for j to coldim(`Z1 Z2`) do print(evaln(re [i,j]) = re[i,j]) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%#reG6$\"\"\"F',&*&&%\"cG6#\"#6F'&%\"aGF,F'F'*&&%\"bG6#\"#7F'&%\"dG6# \"#@F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#reG6$\"\"\"\"\"#,&* &&%\"aG6#\"#6F'&%\"cG6#\"#7F'F'*&&%\"bGF-F'&%\"dGF1F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#reG6$\"\"#\"\"\",$*&&%\"dG6#\"#@F(&%\"bG 6#\"#AF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#reG6$\"\"#F'\"\"! " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1093 1 "\225" }{TEXT -1 45 " values of elements of the imaginar y matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "for i to rowd im(`Z1 Z2`) do for j to coldim(`Z1 Z2`) do print(evaln(im[i,j]) = im [i,j]) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#imG6$\" \"\"F'*&&%\"cG6#\"#6F'&%\"bGF+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %#imG6$\"\"\"\"\"#,&*&&%\"aG6#\"#6F'&%\"dG6#\"#7F'F'*&&%\"bGF-F'&%\"cG F1F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#imG6$\"\"#\"\"\"*&&%\"dG 6#\"#@F(&%\"aG6#\"#AF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#imG6$\" \"#F'\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1090 6 "Step 6" }{TEXT -1 56 ". Construct and display th e real and imaginary matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "R_e := matrix(R_e) : I_m := matrix(I_m) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Real(Z1*Z2) = matrix(R_e) ; `` ; Imag inary(Z1*Z2) = matrix(I_m) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Re alG6#*&%#Z1G\"\"\"%#Z2GF)-%'matrixG6#7$7$,&*&&%\"cG6#\"#6F)&%\"aGF4F)F )*&&%\"bG6#\"#7F)&%\"dG6#\"#@F)!\"\",&*&F6F)&F3F;F)F)*&&F:F4F)&F>F;F)F A7$,$*&F=F)&F:6#\"#AF)FA\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*ImaginaryG6#*&%#Z1G\"\"\"%#Z2GF) -%'matrixG6#7$7$*&&%\"cG6#\"#6F)&%\"bGF3F),&*&&%\"aGF3F)&%\"dG6#\"#7F) F)*&F5F)&F2F=F)F)7$*&&F<6#\"#@F)&F:6#\"#AF)\"\"!" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Thus, the \+ multiplication operation of [" }{TEXT 1094 2 "Z1" }{TEXT -1 7 "] and [ " }{TEXT 1095 2 "Z2" }{TEXT -1 49 "] may be displayed in \"like-in-a-b ook\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "matrix( Z1)*matrix(Z2) = matrix(R_e) + I*matrix(I_m) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7$7$,&&%\"aG6#\"#6\"\"\"*&^#F/F/&%\"bGF- F/F/*&F1F/&F36#\"#7F/7$\"\"!,&&F,6#\"#AF/*&F1F/&F3F " 0 "" {MPLTEXT 1 0 98 "`conj(Z)` := conjugate(Z) : `Conj (Z)` := evalm(conjugate(Z)) : `conj(Z)` = matrix(`Conj(Z)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*conjugateG6#%\"ZG-%'matrixG6#7%7%^ $\"\"\"!\"&^$\"\"#!\"'^$\"\"$\"\"(7%^$\"\"%!\")^$\"\"&\"\"*^$\"\"'!#57 %^$F5!#6^$\"\")!#7^$F<\"#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 642 8 "Method 2" }{TEXT -1 12 ". Using t he " }{TEXT 643 3 "map" }{TEXT -1 5 " and " }{TEXT 644 9 "conjugate" } {TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`Conj(Z)` := map(conjugate, Z) : `conj(Z)` = matrix(`Conj(Z)`) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*conjugateG6#%\"ZG-%'matrixG6#7%7% ^$\"\"\"!\"&^$\"\"#!\"'^$\"\"$\"\"(7%^$\"\"%!\")^$\"\"&\"\"*^$\"\"'!#5 7%^$F5!#6^$\"\")!#7^$F<\"#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 962 8 "Method 3" }{TEXT -1 21 ". Using \+ the function " }{TEXT 964 3 "map" }{TEXT -1 54 " together with the arr ow-type procedure including the " }{TEXT 963 9 "conjugate" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "`Conj(Z)` := map(x->conjugate(x), Z) : `conj(Z)` = matrix(`Conj(Z)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*conjugateG6#%\"ZG-%'matrixG6#7%7%^ $\"\"\"!\"&^$\"\"#!\"'^$\"\"$\"\"(7%^$\"\"%!\")^$\"\"&\"\"*^$\"\"'!#57 %^$F5!#6^$\"\")!#7^$F<\"#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 258 "" 0 "" {TEXT 552 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 553 4 "N.B." }{TEXT -1 49 " A complex matrix, which satisfies the condition" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 577 1 "H" }{TEXT -1 1 "]" } {TEXT 695 14 " = Transp(Conj" }{TEXT -1 1 "[" }{TEXT 687 1 "H" }{TEXT -1 1 "]" }{TEXT 696 1 ")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "is called the " }{TEXT 582 9 "Her mitian" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " } {TEXT 697 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 554