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12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 282 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 283 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 284 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 285 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 2 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 273 "" 0 "" {TEXT 653 39 "MATRICES AND MATRIX OPE RATIONS: Unit 30" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 274 "" 0 "" {TEXT 655 23 "Dr. Wlodzislaw Kostecki" }}{PARA 275 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 276 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 277 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 278 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT 654 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 280 "" 0 "" {TEXT 656 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT 657 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 262 4 "(30)" }{TEXT 481 1 " " }{TEXT 479 27 "Application of th e function" }{TEXT 480 2 " " }{TEXT 257 5 "exp([" }{TEXT 258 1 "A" } {TEXT 259 2 "] " }{TEXT 260 1 "t" }{TEXT 261 1 ")" }{TEXT 263 2 " " } {TEXT 478 27 "in solving matrix equations" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 678 10 "OBJECTIVES" } {TEXT 679 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 680 1 "\225" }{TEXT -1 29 " To introduce the fu nction " }{TEXT 681 5 "exp([" }{TEXT 682 1 "A" }{TEXT 683 2 "] " } {TEXT 684 1 "t" }{TEXT 685 1 ")" }{TEXT -1 62 " for computing the exp onential function of the product of a " }{TEXT 686 15 "constant squar e" }{TEXT -1 16 " matrix and a " }{TEXT 688 4 "real" }{TEXT -1 12 " \+ variable " }{TEXT 687 1 "t" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 689 1 "\225" }{TEXT -1 71 " To exemplify how this operation m ay be performed in alternative ways." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 690 1 "\225" }{TEXT -1 87 " To examine essential properties of matrix multiplication where matrices of the type " }{TEXT 691 5 "exp([" } {TEXT 692 1 "A" }{TEXT 693 2 "] " }{TEXT 694 1 "t" }{TEXT 695 1 ")" } {TEXT -1 15 " are involved." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 713 1 " \225" }{TEXT -1 90 " To provide a brief introduction to, and necessar y formula for, solving matrix equations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 719 1 "\225" }{TEXT -1 84 " To state a condition necessary for \+ the solution to a matrix equation to be unique." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 714 1 "\225" }{TEXT -1 89 " To provide a step-by-step exam ple of solving a matrix equation and verify the solution." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 715 1 "\225" }{TEXT -1 104 " To provide a brief introduction to, and necessary formulae for, solving matrix different ial equations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 718 1 "\225" }{TEXT -1 102 " To provide a step-by-step example of solving a matrix differ ential equation and verify the solution." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 742 1 "\225" }{TEXT -1 81 " To provide a universal method for c learing fractions from matrix elements with " }{TEXT 743 5 "Maple" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "restart : with(linalg, coldim, di ag, eigenvals, equal, exponential, multiply, rowdim) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "T he function " }{TEXT 264 5 "exp([" }{TEXT 265 1 "A" }{TEXT 266 2 "] \+ " }{TEXT 267 1 "t" }{TEXT 268 1 ")" }{TEXT -1 9 " for a " }{TEXT 269 15 "constant square" }{TEXT -1 10 " matrix [" }{TEXT 270 1 "A" } {TEXT -1 7 "] and " }{TEXT 272 4 "real" }{TEXT -1 12 " variable " } {TEXT 271 1 "t" }{TEXT -1 28 " is of special importance." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "T he function was used in the form " }{TEXT 705 4 "exp(" }{TEXT -1 1 "[ " }{TEXT 701 1 "A" }{TEXT 703 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 702 1 ")" }{TEXT -1 1 "]" }{TEXT 704 1 ")" }{TEXT -1 51 " in Un it (24) where elements of a square matrix [" }{TEXT 710 1 "A" }{TEXT 712 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 711 1 ")" }{TEXT -1 61 " ] contained only the proportional terms of linear function " } {XPPEDIT 18 0 "a*t+b" "6#,&*&%\"aG\"\"\"%\"tGF&F&%\"bGF&" }{TEXT 709 1 "," }{TEXT -1 46 " for computing the exponential function of [" } {TEXT 706 1 "A" }{TEXT 708 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 707 1 ")" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "However, the major application of \+ the function " }{TEXT 696 5 "exp([" }{TEXT 697 1 "A" }{TEXT 698 2 "] \+ " }{TEXT 699 1 "t" }{TEXT 700 1 ")" }{TEXT -1 67 " is in solving matr ix equations and matrix differential equations." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The matrix f unction " }{TEXT 273 5 "exp([" }{TEXT 274 1 "A" }{TEXT 275 2 "] " } {TEXT 276 1 "t" }{TEXT 277 1 ")" }{TEXT -1 31 " may be computed by se tting [" }{TEXT 278 1 "B" }{TEXT -1 1 "]" }{TEXT 602 3 " = " }{TEXT -1 1 "[" }{TEXT 279 1 "A" }{TEXT -1 2 "] " }{TEXT 280 1 "t" }{TEXT -1 24 " and then calculating " }{TEXT 281 5 "exp([" }{TEXT 282 1 "B" } {TEXT 283 3 "])," }{TEXT -1 6 " or " }{TEXT 472 5 "exp([" }{TEXT 473 1 "A" }{TEXT 474 2 "] " }{TEXT 475 1 "t" }{TEXT 476 1 ")" }{TEXT -1 38 " may be computed directly, using the " }{TEXT 659 11 "exponent ial" }{TEXT -1 10 " function." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "A few examples of computation of the matrix function " }{TEXT 284 5 "exp([" }{TEXT 285 1 "A" } {TEXT 286 2 "] " }{TEXT 287 1 "t" }{TEXT 288 1 ")" }{TEXT -1 35 " for different constant matrices [" }{TEXT 289 1 "A" }{TEXT -1 9 "] follow ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 9 "Example 1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 293 5 "exp([" } {TEXT 294 1 "A" }{TEXT 295 2 "] " }{TEXT 296 1 "t" }{TEXT 297 1 ")" } {TEXT -1 9 " for a " }{TEXT 603 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 291 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 604 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 292 1 "A" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := matrix(2, 2, [0, 1, -1 , 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%' matrixG6#7$7$\"\"!\"\"\"7$!\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "B := evalm(A * t) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6 #7$7$\"\"!%\"tG7$,$F+!\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The matrix function " }{TEXT 298 5 "exp([" }{TEXT 299 1 "A" }{TEXT 300 2 "] " }{TEXT 301 1 "t" } {TEXT 302 1 ")" }{TEXT -1 26 " is the following matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`exp(B)` := exponential(B) : exp( A*t) = matrix(`exp(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6 #*&%\"AG\"\"\"%\"tGF)-%'matrixG6#7$7$-%$cosG6#F*-%$sinGF27$,$F3!\"\"F0 " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The same result may be obtained in " }{TEXT 660 5 "Maple " }{TEXT -1 2 " " }{TEXT 477 8 "directly" }{TEXT -1 12 " using the \+ " }{TEXT 662 11 "exponential" }{TEXT -1 38 " function and scalar multi plication [" }{TEXT 663 1 "A" }{TEXT -1 2 "] " }{TEXT 664 1 "t" } {TEXT 675 1 "," }{TEXT -1 6 " viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`exp(At)` := exponential(A * t) : exp(A*t) = matrix (`exp(At)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"AG\"\" \"%\"tGF)-%'matrixG6#7$7$-%$cosG6#F*-%$sinGF27$,$F3!\"\"F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "o r using the " }{TEXT 665 11 "exponential" }{TEXT -1 15 " function and \+ " }{TEXT 661 1 "t" }{TEXT -1 29 " as a scalar parameter, viz." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`exp(At)` := exponential(A, \+ t) : exp(A*t) = matrix(`exp(At)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"AG\"\"\"%\"tGF)-%'matrixG6#7$7$-%$cosG6#F*-%$sinGF27 $,$F3!\"\"F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 282 "" 0 "" {TEXT 673 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 672 4 "N.B." }{TEXT -1 23 " The eig envalues of [" }{TEXT 668 1 "B" }{TEXT -1 1 "]" }{TEXT 671 3 " = " } {TEXT -1 1 "[" }{TEXT 669 1 "A" }{TEXT -1 2 "] " }{TEXT 670 1 "t" } {TEXT -1 26 " are the eigenvalues of [" }{TEXT 666 1 "A" }{TEXT -1 17 "] multiplied by " }{TEXT 667 1 "t" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : ch ar_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+ch ar_rootsG6#%\"AG6$^#\"\"\"^#!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(B) := eige nvals(B) : char_roots(B) = charroots(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"BG6$*&^#\"\"\"F+%\"tGF+*&^#!\"\"F+F ,F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 674 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 303 9 "Example 2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " } {TEXT 306 5 "exp([" }{TEXT 307 1 "U" }{TEXT 308 2 "] " }{TEXT 309 1 "t " }{TEXT 310 1 ")" }{TEXT -1 11 " for the " }{TEXT 605 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 304 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 606 1 ")" }{TEXT -1 15 " unit matrix [" }{TEXT 305 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U := \+ diag(1, 1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" UG-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`exp(Ut)` := expo nential(U, t) : exp(U*t) = matrix(`exp(Ut)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"UG\"\"\"%\"tGF)-%'matrixG6#7$7$-F%6#F*\" \"!7$F2F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 311 9 "Example 3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 314 5 "exp([ " }{TEXT 315 1 "A" }{TEXT 316 2 "] " }{TEXT 317 1 "t" }{TEXT 318 1 ") " }{TEXT -1 9 " for a " }{TEXT 607 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"# " }{TEXT 312 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 608 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 313 1 "A" }{TEXT -1 10 "] given as" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := matrix(2, 2, [1, 1, - 1, 1]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-% 'matrixG6#7$7$\"\"\"F*7$!\"\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`exp(At)` := exponenti al(A, t) : exp(A*t) = matrix(`exp(At)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"AG\"\"\"%\"tGF)-%'matrixG6#7$7$*&-F%6#F* F)-%$cosGF2F)*&F1F)-%$sinGF2F)7$,$F5!\"\"F0" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 319 9 "Example 4" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 322 5 "exp([" }{TEXT 323 1 "A" }{TEXT 324 2 "] " }{TEXT 325 1 "t" }{TEXT 326 1 ")" }{TEXT -1 8 " for a " }{TEXT 609 2 " (" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 320 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 610 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 321 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := matrix(2, 2, [0, -1, 2, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"!!\"\" 7$\"\"#F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 65 "`exp(At)` := exponential(A, t) : exp(A*t) \+ = matrix(`exp(At)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&% \"AG\"\"\"%\"tGF)-%'matrixG6#7$7$-%$cosG6#*&-%%sqrtG6#\"\"#F)F*F),$*&F 4F)-%$sinGF2F)#!\"\"F77$F9F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 327 9 "Example 5" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 329 5 "exp([" }{TEXT 330 1 "A" }{TEXT 331 2 "] " }{TEXT 332 1 " t" }{TEXT 333 1 ")" }{TEXT -1 9 " for a " }{TEXT 611 1 "(" } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 334 3 " \327 " }{XPPEDIT 18 0 "4" " 6#\"\"%" }{TEXT 612 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 328 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " A := matrix(4, 4, [1, 2, 3, 4, 0, 1, 2, 3, 0, 0, 1, 2, 0, 0, 0, 1]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6 #7&7&\"\"\"\"\"#\"\"$\"\"%7&\"\"!F*F+F,7&F/F/F*F+7&F/F/F/F*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`exp(At)` := exponential(A, t) : exp(A*t) = matrix( `exp(At)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"AG\"\" \"%\"tGF)-%'matrixG6#7&7&-F%6#F*,$*&F*F)F0F)\"\"#,&*&)F*F4F)F0F)F4*(\" \"$F)F*F)F0F)F),(*&)F*F9F)F0F)#\"\"%F9*(\"\"'F)F7F)F0F)F)*(F>F)F*F)F0F )F)7&\"\"!F0F2F57&FCFCF0F27&FCFCFCF0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The above matrix may be displayed in a more compact form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "`exp(At)` := map(collect, `exp(At)`, exp(t)) : exp( A*t) = matrix(`exp(At)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG 6#*&%\"AG\"\"\"%\"tGF)-%'matrixG6#7&7&-F%6#F*,$*&F*F)F0F)\"\"#*&,&*$)F *F4F)F4*&\"\"$F)F*F)F)F)F0F)*&,(*$)F*F:F)#\"\"%F:*&\"\"'F)F8F)F)*&F@F) F*F)F)F)F0F)7&\"\"!F0F2F57&FEFEF0F27&FEFEFEF0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 342 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 336 4 "N.B." }{TEXT -1 49 " Computation of the the matrix exponential of [" }{TEXT 335 1 "A" }{TEXT -1 2 "] " }{TEXT 337 1 "t" }{TEXT -1 39 " depends on finding eigenvalues for [" }{TEXT 338 1 "A" }{TEXT -1 2 "] " }{TEXT 339 1 "t" }{TEXT 340 1 "." }{TEXT -1 36 " A symbolic result can be achieved " }{TEXT 341 4 "only" }{TEXT -1 38 " if symbol ic eigenvalues can be found." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 343 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 347 4 "N.B." } {TEXT -1 20 " If two matrices, [" }{TEXT 344 1 "A" }{TEXT -1 7 "] and [" }{TEXT 345 1 "B" }{TEXT -1 16 "], satisfy the " }{TEXT 346 11 "co mmutative" }{TEXT -1 29 " law of multiplication, then" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 348 5 "exp([" }{TEXT 349 1 "A" }{TEXT 350 2 "] " }{TEXT 351 1 "t" }{TEXT 352 7 ") exp([" }{TEXT 356 1 "B" }{TEXT 353 2 "] " }{TEXT 354 1 "t" }{TEXT 355 9 ") = exp[(" }{TEXT -1 1 "[" } {TEXT 613 1 "A" }{TEXT -1 1 "]" }{TEXT 358 4 " + [" }{TEXT 359 1 "B" } {TEXT -1 1 "]" }{TEXT 360 2 ") " }{TEXT 361 1 "t" }{TEXT 362 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 363 31 "Exemplarily, consider the two " }{TEXT 357 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 366 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 367 1 ")" }{TEXT 614 2 " " }{TEXT 370 9 "commuting" }{TEXT 371 12 " matrices [" }{TEXT 364 1 "A" }{TEXT 368 7 "] and [" }{TEXT 365 1 "B" }{TEXT 369 22 "] of Unit (4) given as" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 107 "A := matrix(2, 2, [3, 4, 2, 3]) : A = matri x(A) ; B := matrix(2, 2, [3, -4, -2, 3]) : B = matrix(B) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"$\"\"%7$\"\" #F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\"\"$!\"% 7$!\"#F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 17 "(a) The product " }{TEXT 372 5 "exp([" }{TEXT 373 1 "A" }{TEXT 374 2 "] " }{TEXT 375 1 "t" }{TEXT 376 7 ") exp([" } {TEXT 380 1 "B" }{TEXT 377 2 "] " }{TEXT 378 1 "t" }{TEXT 379 1 ")" } {TEXT -1 20 " is the following " }{TEXT 630 1 "(" }{XPPEDIT 18 0 "2 " "6#\"\"#" }{TEXT 629 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 631 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`exp(At) + exp(Bt)` := multiply(exponential(A, t), ex ponential(B, t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`exp( At) + exp(Bt)` := map(simplify, `exp(At) + exp(Bt)`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "exp(A*t) * exp(B*t) = matrix(`exp(A t) + exp(Bt)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#*&%\"A G\"\"\"%\"tGF*F*-F&6#*&%\"BGF*F+F*F*-%'matrixG6#7$7$-F&6#,$F+\"\"'\"\" !7$F9F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(b) The exponential " }{TEXT 615 5 "exp[(" }{TEXT -1 1 "[" }{TEXT 621 1 "A" }{TEXT -1 1 "]" }{TEXT 616 4 " + [" }{TEXT 617 1 "B" }{TEXT -1 1 "]" }{TEXT 618 2 ") " }{TEXT 619 1 "t" }{TEXT 620 1 "]" }{TEXT -1 20 " is the following " }{TEXT 633 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 632 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 634 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "`exp(A + Bt)` := exponential((evalm(A + B)), \+ t) : exp((A+B)*t) = matrix(`exp(A + Bt)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&,&%\"AG\"\"\"%\"BGF*F*%\"tGF*-%'matrixG6#7$ 7$-F%6#,$F,\"\"'\"\"!7$F6F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Both matrices of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 " " 0 "" {TEXT 381 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 383 4 "N.B." }{TEXT -1 11 " For any " } {TEXT 399 15 "constant square" }{TEXT -1 10 " matrix [" }{TEXT 382 1 "A" }{TEXT -1 7 "] and " }{TEXT 414 4 "real" }{TEXT -1 13 " variable s " }{TEXT 400 1 "t" }{TEXT -1 7 " and " }{TEXT 401 1 "s" }{TEXT 402 1 "," }}}{EXCHG {PARA 262 "" 0 "" {TEXT 384 5 "exp([" }{TEXT 385 1 "A" }{TEXT 386 2 "] " }{TEXT 387 1 "t" }{TEXT 388 8 ") exp(\226[" } {TEXT 394 1 "A" }{TEXT 389 2 "] " }{TEXT 393 1 "s" }{TEXT 390 9 ") = e xp\{[" }{TEXT 391 1 "A" }{TEXT 392 3 "] (" }{TEXT 395 1 "t" }{TEXT 396 3 " \226 " }{TEXT 397 1 "s" }{TEXT 398 3 ")]\}" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Exemplari ly, consider the above matrix [" }{TEXT 403 1 "A" }{TEXT -1 2 "]." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(a) The product " }{TEXT 404 5 "exp([" }{TEXT 405 1 "A" } {TEXT 406 2 "] " }{TEXT 407 1 "t" }{TEXT 408 8 ") exp(\226[" }{TEXT 412 1 "A" }{TEXT 409 2 "] " }{TEXT 411 1 "s" }{TEXT 410 1 ")" }{TEXT -1 26 " is the following matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`exp(At) exp(-As)` := multiply(exponential(A, t), exp onential(-A, s)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`exp( At) exp(-As)` := map(combine, map(expand, `exp(At) exp(-As)`)) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "exp(A*t) * exp(-A*s) = matri x(`exp(At) exp(-As)`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6 #*&%\"AG\"\"\"%\"tGF*F*-F&6#,$*&F)F*%\"sGF*!\"\"F*-%'matrixG6#7$7$,&-F &6#,*F+\"\"$*(\"\"#F*-%%sqrtG6#F=F*F+F*F1*&F;F*F0F*F1*(F=F*F0F*F>F*F*# F*F=*&FCF*-F&6#,*F+F;*(F=F*F>F*F+F*F**&F;F*F0F*F1*(F=F*F0F*F>F*F1F*F*, &*&F>F*FEF*FC*&#F*F=F**&F>F*F8F*F*F17$,&FO#F1\"\"%*(#F*FSF*F>F*FEF*F*F 7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(b) The exponential " }{TEXT 622 5 "exp\{[" }{TEXT 623 1 "A" }{TEXT 624 3 "] (" }{TEXT 625 1 "t" }{TEXT 626 3 " \226 " } {TEXT 627 1 "s" }{TEXT 628 3 ")]\}" }{TEXT -1 26 " is the following m atrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "`exp(A(t-s))` := \+ exponential(evalm(A*(t-s))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`exp(A(t-s))` := map(combine, map(expand, `exp(A(t-s))`)) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "exp(A*(t-s)) = matrix(`exp(A (t-s))`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"AG\"\"\", &%\"tGF)%\"sG!\"\"F)-%'matrixG6#7$7$,&-F%6#,*F+\"\"$*(\"\"#F)-%%sqrtG6 #F9F)F+F)F-*&F7F)F,F)F-*(F9F)F,F)F:F)F)#F)F9*&F?F)-F%6#,*F+F7*(F9F)F:F )F+F)F)*&F7F)F,F)F-*(F9F)F,F)F:F)F-F)F),&*&F:F)FAF)F?*&#F)F9F)*&F:F)F4 F)F)F-7$,&FK#F-\"\"%*(#F)FOF)F:F)FAF)F)F3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 720 5 "equal" }{TEXT -1 64 " function applied to the resultant matrices of (a) and (b), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "equal (`exp(At) exp(-As)`, `exp(A(t-s))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "returns the " }{TEXT 721 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 47 ", which verifies that both matrices are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 263 "" 0 "" {TEXT 413 5 "* * *" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The two ma jor application domains of the function " }{TEXT 415 5 "exp([" } {TEXT 416 1 "A" }{TEXT 417 2 "] " }{TEXT 418 1 "t" }{TEXT 419 1 ")" } {TEXT -1 40 " are illustrated hereunder in Sections " }{TEXT 676 1 "A " }{TEXT -1 5 " and " }{TEXT 677 1 "B" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 420 3 "A . " }{TEXT 421 28 "Solution of matrix equations" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The equation " }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 1 "[" }{TEXT 422 1 "A" }{TEXT -1 3 "] [" }{TEXT 423 1 "X" }{TEXT -1 1 "]" }{TEXT 635 3 " + " }{TEXT -1 1 "[" }{TEXT 424 1 "X" }{TEXT -1 3 "] [" }{TEXT 425 1 "B" }{TEXT -1 1 "]" }{TEXT 636 3 " = " }{TEXT -1 1 "[" }{TEXT 426 1 "C" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where [" }{TEXT 427 1 "A" }{TEXT -1 4 "], [" }{TEXT 428 1 "B" }{TEXT -1 8 "], and [" }{TEXT 429 1 "C" }{TEXT -1 10 "] deno te " }{TEXT 637 8 "constant" }{TEXT -1 60 " square matrices of the s ame order, has a unique solution [" }{TEXT 430 1 "X" }{TEXT -1 18 "] i f and only if [" }{TEXT 431 1 "A" }{TEXT -1 7 "] and [" }{TEXT 432 1 " B" }{TEXT -1 65 "] have no eigenvalues in common. This unique solution is given by" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 1 "[" }{TEXT 433 1 "X" }{TEXT -1 1 "]" }{TEXT 434 5 " = \226 " }{XPPEDIT 18 0 "Int(exp(A* t)*C*exp(B*t), t=0..infinity)" "6#-%$IntG6$*(-%$expG6#*&%\"AG\"\"\"%\" tGF,F,%\"CGF,-F(6#*&%\"BGF,F-F,F,/F-;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "p rovided the improper integral exists." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Exemplarily, solv e the matrix equation [" }{TEXT 510 1 "A" }{TEXT -1 3 "] [" }{TEXT 511 1 "X" }{TEXT -1 1 "]" }{TEXT 638 3 " + " }{TEXT -1 1 "[" }{TEXT 512 1 "X" }{TEXT -1 3 "] [" }{TEXT 513 1 "B" }{TEXT -1 1 "]" }{TEXT 639 3 " = " }{TEXT -1 1 "[" }{TEXT 514 1 "C" }{TEXT -1 8 "] for [" } {TEXT 521 1 "X" }{TEXT -1 5 "] if " }{TEXT 515 1 " " }{TEXT 640 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 516 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 517 1 ")" }{TEXT 641 11 " matrices " }{TEXT -1 1 "[ " }{TEXT 518 1 "A" }{TEXT -1 4 "], [" }{TEXT 519 1 "B" }{TEXT -1 8 "], and [" }{TEXT 520 1 "C" }{TEXT -1 14 "] are given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "A := matrix(2, 2, [-6, 0, 0, -8]) \+ : B := matrix(2, 2, [3, 2, 4, 1]) : C := matrix(2, 2, [2, 0, -1, 0 ]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A = matrix(A) ; B = matrix(B) ; C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"AG-%'matrixG6#7$7$!\"'\"\"!7$F+!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\"\"$\"\"#7$\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7$7$\"\"#\"\"!7$!\"\"F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 435 6 "S tep 1" }{TEXT -1 36 ". Find the eigenvalues of matrices [" }{TEXT 438 1 "A" }{TEXT -1 7 "] and [" }{TEXT 439 1 "B" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 436 1 "\225" }{TEXT -1 14 " for matrix [" }{TEXT 437 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eig envals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$!\"'!\")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 440 1 "\225" } {TEXT -1 14 " for matrix [" }{TEXT 441 1 "B" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(B) := eigenvals(B) : char_roots(B) = charroots(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%+char_rootsG6#%\"BG6$\"\"&!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "The two matrices have n o eigenvalues in common. Thus, the equation has a unique solution [" } {TEXT 642 1 "X" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 442 6 "Step 2" }{TEXT -1 42 ". Dete rmine the exponential functions of [" }{TEXT 443 1 "A" }{TEXT -1 7 "] \+ and [" }{TEXT 444 1 "B" }{TEXT -1 15 "] in variable " }{TEXT 445 1 "t " }{TEXT 446 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 447 1 "\225" }{TEXT -1 14 " for matrix [" } {TEXT 448 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`exp(At)` := exponential(A, t) : exp(A*t) = matrix(`exp(At)` ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"AG\"\"\"%\"tGF)- %'matrixG6#7$7$-F%6#,$F*!\"'\"\"!7$F4-F%6#,$F*!\")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 449 1 "\225" } {TEXT -1 14 " for matrix [" }{TEXT 450 1 "B" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`exp(Bt)` := map(sort, expon ential(B, t)) : exp(B*t) = matrix(`exp(Bt)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"BG\"\"\"%\"tGF)-%'matrixG6#7$7$,&-F%6#,$ F*!\"\"#F)\"\"$*&#\"\"#F6F)-F%6#,$F*\"\"&F)F),&F1#F4F6*&F5F)F:F)F)7$,& F1#!\"#F6*&F8F)F:F)F),&F1F8*&F5F)F:F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 451 6 "Step 3" }{TEXT -1 34 ". Determine the matrix integrand " }{TEXT 452 5 "exp([" } {TEXT 453 1 "A" }{TEXT 454 1 "]" }{TEXT 459 2 " t" }{TEXT 460 3 ") [" }{TEXT 457 1 "C" }{TEXT 455 7 "] exp([" }{TEXT 458 1 "B" }{TEXT 456 1 "]" }{TEXT 461 2 " t" }{TEXT 462 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "M(t) := evalm(`exp(At)` &* C &* `exp(Bt)`) : exp(A* t) * C * exp(B*t) = M(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%$ex pG6#*&%\"AG\"\"\"%\"tGF*F*%\"CGF*-F&6#*&%\"BGF*F+F*F*-%'matrixG6#7$7$, $*&-F&6#,$F+!\"'F*,&-F&6#,$F+!\"\"#F*\"\"$*&#\"\"#FBF*-F&6#,$F+\"\"&F* F*F*FE,$*&F8F*,&F=#F@FB*&FAF*FFF*F*F*FE7$,$*&-F&6#,$F+!\")F*F " 0 "" {MPLTEXT 1 0 88 "M(t) := map(combine, map(expand, M(t), power), exp) : exp(A*t) * C * exp(B*t) = M(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%$expG6# *&%\"AG\"\"\"%\"tGF*F*%\"CGF*-F&6#*&%\"BGF*F+F*F*-%'matrixG6#7$7$,&-F& 6#,$F+!\"(#\"\"#\"\"$*&#\"\"%F=F*-F&6#,$F+!\"\"F*F*,&F7#!\"#F=*&F;F*FA F*F*7$,&-F&6#,$F+!\"*#FDF=*&#F " 0 "" {MPLTEXT 1 0 125 "`int(M(t))` := evalm(map(-int, M(t), t=0..infinity)) : -Int(exp(A*t) * C * exp(B*t), t=0..infinity) = matrix(`int(M(t))` ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$IntG6$*(-%$expG6#*&%\"AG \"\"\"%\"tGF.F.%\"CGF.-F*6#*&%\"BGF.F/F.F./F/;\"\"!%)infinityG!\"\"-%' matrixG6#7$7$#!#5\"\"(#!\"%FA7$#FA\"#F#\"\"#FF" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 539 6 "Step 5" } {TEXT -1 30 ". Obtain the solution matrix [" }{TEXT 538 1 "X" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "X := matrix(`in t(M(t))`) : X = matrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"X G-%'matrixG6#7$7$#!#5\"\"(#!\"%F,7$#F,\"#F#\"\"#F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 725 21 "Clearing of fractions" }{TEXT -1 253 " may be applied to the elements of this matrix by multiplying them by the least common multiple of their deno minators and dividing the new matrix by the same. If this to be done m anually, it is intuitive and simple. If the task is to be performed wi th " }{TEXT 724 5 "Maple" }{TEXT -1 76 ", the procedure is quite elabo rate and requires several operations as shown." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "(a) Create a list of the elements of [" }{TEXT 726 1 "X" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L := convert(convert(X, set) , list) : elements(X) = L ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)e lementsG6#%\"XG7&#\"\"#\"#F#!\"%\"\"(#!#5F.#F.F+" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "(b) Extrac t denominators of the elements of the list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "for i to nops(L) do den(el[i]('L')) := denom(op( i, L)) : print(evaln(denominator(element[i]('L'))) = den(el[i]('L')) ) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(el ementG6#\"\"\"6#%\"LG\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denom inatorG6#-&%(elementG6#\"\"#6#%\"LG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6#\"\"$6#%\"LG\"\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6#\"\"%6#%\"LG\"#F " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "(c) Create a sequence of denominators of the elements of \+ the list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "`seq(den(el(L )))` := [seq(den(el[n]('L')), n=1..nops(L))] : den(el('L')) := op(`s eq(den(el(L)))`) : denominators(elements(X)) = den(el('L')) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-denominatorsG6#-%)elementsG6#%\"XG 6&\"#F\"\"(F-F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "(d) Find the least common multiple of den ominators of the elements of the list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "LCM := lcm(den(el('L'))) : least_common_multiple(de nominators(X)) = LCM ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%6least_co mmon_multipleG6#-%-denominatorsG6#%\"XG\"$*=" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "(e) Multiply th e elements of [" }{TEXT 727 1 "X" }{TEXT -1 91 "] by the least common \+ multiple of their denominators and divide the new matrix by the same: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "X := evalm(X*LCM)/LCM \+ : 'X' = X ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"XG,$-%'matrixG6#7$ 7$!$q#!$3\"7$\"#\\\"#9#\"\"\"\"$*=" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 464 6 "Step 6" }{TEXT -1 52 ". Verify the solution by computing the sum matrix [" }{TEXT 465 1 "A" }{TEXT -1 3 "] [" }{TEXT 466 1 "X" }{TEXT -1 1 "]" }{TEXT 643 3 " + " }{TEXT -1 1 "[" }{TEXT 467 1 "X" }{TEXT -1 3 "] [" }{TEXT 468 1 "B" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "X := eval m(X) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`AX + XB` := eval m(A &* X + X &* B) : A*X + X*B = matrix(`AX + XB`) ; X := 'X' :" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"AG\"\"\"%\"XGF'F'*&F(F'%\"BGF 'F'-%'matrixG6#7$7$\"\"#\"\"!7$!\"\"F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The sum matrix [ " }{TEXT 557 1 "A" }{TEXT -1 3 "] [" }{TEXT 558 1 "X" }{TEXT -1 1 "]" }{TEXT 644 3 " + " }{TEXT -1 1 "[" }{TEXT 559 1 "X" }{TEXT -1 3 "] [" }{TEXT 560 1 "B" }{TEXT -1 23 "] is equal to matrix [" }{TEXT 469 1 " C" }{TEXT -1 31 "], which verifies the solution." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 284 "" 0 "" {TEXT 716 5 "* * *" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 470 3 "B. " }{TEXT 471 41 "Solution of matrix differential equat ions" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 60 "The initial-value matrix differential inhomogeneous equ ation" }}}{EXCHG {PARA 266 "" 0 "" {XPPEDIT 18 0 "Diff(X(t),t)" "6#-%% DiffG6$-%\"XG6#%\"tGF)" }{TEXT 645 3 " = " }{TEXT -1 1 "[" }{TEXT 485 1 "A" }{TEXT -1 2 "][" }{TEXT 486 1 "X" }{TEXT 483 1 "(" }{TEXT 482 1 "t" }{TEXT 484 1 ")" }{TEXT -1 2 "] " }{TEXT 487 1 "+" }{TEXT -1 2 " [ " }{TEXT 491 1 "F" }{TEXT 488 1 "(" }{TEXT 489 1 "t" }{TEXT 490 1 ")" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "with initial condition" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 1 "[" }{TEXT 495 1 "X" }{TEXT 492 1 "(" } {XPPEDIT 18 0 "t[0]" "6#&%\"tG6#\"\"!" }{TEXT 493 1 ")" }{TEXT -1 1 "] " }{TEXT 646 3 " = " }{TEXT -1 1 "[" }{TEXT 494 1 "C" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "has the solution" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 1 "[" }{TEXT 499 1 "X" }{TEXT 497 1 "(" }{TEXT 496 1 "t" }{TEXT 498 1 ") " }{TEXT -1 1 "]" }{TEXT 501 3 " = " }{XPPEDIT 18 0 "exp(A*(t-t[0]))" "6#-%$expG6#*&%\"AG\"\"\",&%\"tGF(&F*6#\"\"!!\"\"F(" }{TEXT -1 2 " [" }{TEXT 500 1 "C" }{TEXT -1 1 "]" }{TEXT 502 3 " + " }{XPPEDIT 18 0 "ex p(A*t) * Int(exp(-A*s)*F(s),s=t[0]..t)" "6#*&-%$expG6#*&%\"AG\"\"\"%\" tGF)F)-%$IntG6$*&-F%6#,$*&F(F)%\"sGF)!\"\"F)-%\"FG6#F3F)/F3;&F*6#\"\"! F*F)" }{TEXT 647 15 " (1)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "or, equivalently," }}} {EXCHG {PARA 269 "" 0 "" {TEXT -1 1 "[" }{TEXT 506 1 "X" }{TEXT 504 1 "(" }{TEXT 503 1 "t" }{TEXT 505 1 ")" }{TEXT -1 1 "]" }{TEXT 508 3 " = " }{XPPEDIT 18 0 "exp(A*(t-t[0]))" "6#-%$expG6#*&%\"AG\"\"\",&%\"tGF( &F*6#\"\"!!\"\"F(" }{TEXT -1 2 " [" }{TEXT 507 1 "C" }{TEXT -1 1 "]" } {TEXT 509 3 " + " }{XPPEDIT 18 0 "Int(exp(A*(t-s))*F(s),s=t[0]..t)" "6 #-%$IntG6$*&-%$expG6#*&%\"AG\"\"\",&%\"tGF,%\"sG!\"\"F,F,-%\"FG6#F/F,/ F/;&F.6#\"\"!F." }{TEXT 648 20 " (2)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 561 7 "NOTE 1: " }{TEXT -1 52 " If the differential equation is homogeneous, i.e.," }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 1 "[" }{TEXT 565 1 "F" }{TEXT 562 1 "(" }{TEXT 563 1 "t" }{TEXT 564 1 ")" }{TEXT -1 1 "]" }{TEXT 649 3 " = " }{TEXT -1 1 "[" }{TEXT 566 1 "0" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "then formulae (1) and (2) reduce to" }}}{EXCHG {PARA 271 "" 0 " " {TEXT -1 1 "[" }{TEXT 570 1 "X" }{TEXT 568 1 "(" }{TEXT 567 1 "t" } {TEXT 569 1 ")" }{TEXT -1 1 "]" }{TEXT 572 3 " = " }{XPPEDIT 18 0 "exp (A*(t-t[0]))" "6#-%$expG6#*&%\"AG\"\"\",&%\"tGF(&F*6#\"\"!!\"\"F(" } {TEXT -1 2 " [" }{TEXT 571 1 "C" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 573 7 "NOTE 2:" } {TEXT -1 13 " Obtaining [" }{TEXT 577 1 "X" }{TEXT 575 1 "(" }{TEXT 574 1 "t" }{TEXT 576 1 ")" }{TEXT -1 189 "] from formula (2) requires \+ one operation step less than using formula (1). However, the integrals arising in formula (2) are generally more difficult to evaluate than \+ those in formula (1)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "As an example, use formula (1) to \+ find the solution [" }{TEXT 581 1 "X" }{TEXT 579 1 "(" }{TEXT 578 1 "t " }{TEXT 580 1 ")" }{TEXT -1 32 "] to the differential equation " } {XPPEDIT 18 0 "Diff(X(t),t)" "6#-%%DiffG6$-%\"XG6#%\"tGF)" }{TEXT 650 3 " = " }{TEXT -1 1 "[" }{TEXT 585 1 "A" }{TEXT -1 2 "][" }{TEXT 586 1 "X" }{TEXT 583 1 "(" }{TEXT 582 1 "t" }{TEXT 584 1 ")" }{TEXT -1 2 " ] " }{TEXT 587 1 "+" }{TEXT -1 2 " [" }{TEXT 591 1 "F" }{TEXT 588 1 "( " }{TEXT 589 1 "t" }{TEXT 590 1 ")" }{TEXT -1 24 "] with initial valu e [" }{TEXT 595 1 "X" }{TEXT 592 1 "(" }{XPPEDIT 18 0 "t[0]" "6#&%\"t G6#\"\"!" }{TEXT 593 1 ")" }{TEXT -1 1 "]" }{TEXT 651 3 " = " }{TEXT -1 1 "[" }{TEXT 594 1 "C" }{TEXT -1 7 "] if " }{XPPEDIT 18 0 "t[0]=0 " "6#/&%\"tG6#\"\"!F'" }{TEXT -1 18 ", when matrices [" }{TEXT 600 1 "A" }{TEXT -1 4 "], [" }{TEXT 599 1 "F" }{TEXT 596 1 "(" }{TEXT 597 1 "t" }{TEXT 598 1 ")" }{TEXT -1 8 "], and [" }{TEXT 601 1 "C" }{TEXT -1 14 "] are given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "A := matrix(2, 2, [0, 1, 8, -2]) : F(t) := matrix(2, 1, [0, exp(t)]) \+ : C := matrix(2, 1, [1, -4]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A = matrix(A) ; 'F(t)' = F(t) ; C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"!\"\"\"7$ \"\")!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"tG-%'matrixG6 #7$7#\"\"!7#-%$expGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matr ixG6#7$7#\"\"\"7#!\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 522 6 "Step 1" }{TEXT -1 41 ". Determine \+ the exponential function of [" }{TEXT 523 1 "A" }{TEXT -1 15 "] in var iable " }{XPPEDIT 18 0 "t-t[0]" "6#,&%\"tG\"\"\"&F$6#\"\"!!\"\"" } {TEXT 524 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`exp(A(t -to))` := subs(t[0]=0, evalm(exponential(A, t) &* exponential(-A, t[0] ))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "exp(A*(t-t[0])) = \+ matrix(`exp(A(t-to))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6# *&%\"AG\"\"\",&%\"tGF)&F+6#\"\"!!\"\"F)-%'matrixG6#7$7$*&,&-F%6#,$F+! \"%#F)\"\"$*&#\"\"#F*&F;F)F@F)F)F )FCF)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "After some mathematical operations, the above matrix s implifies to the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`e xp(A(t-to))` := map(sort, map(combine, map(expand, `exp(A(t-to))`, pow er), exp)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "exp(A*(t-t[ 0])) = matrix(`exp(A(t-to))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $expG6#*&%\"AG\"\"\",&%\"tGF)&F+6#\"\"!!\"\"F)-%'matrixG6#7$7$,&-F%6#, $F+\"\"##F9\"\"$*&#F)F;F)-F%6#,$F+!\"%F)F),&F6#F)\"\"'*&#F)FDF)F>F)F/7 $,&F6#\"\"%F;*&#FJF;F)F>F)F/,&F6F=*&F:F)F>F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 526 6 "Step 2" } {TEXT -1 32 ". Determine the product matrix " }{XPPEDIT 18 0 "exp(A*( t-t[0]))" "6#-%$expG6#*&%\"AG\"\"\",&%\"tGF(&F*6#\"\"!!\"\"F(" }{TEXT -1 2 " [" }{TEXT 525 1 "C" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 100 "`exp(A(t-to)) C` := multiply(`exp(A(t-to))`, C) : exp(A*(t-t[0])) * C = matrix(`exp(A(t-to)) C`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&-%$expG6#*&%\"AG\"\"\",&%\"tGF*&F,6#\"\"!!\"\"F*F* %\"CGF*-%'matrixG6#7$7#-F&6#,$F,!\"%7#,$F7F:" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 528 6 "Step 3" }{TEXT -1 41 ". Determine the exponential function of [" }{TEXT 527 1 "A" } {TEXT -1 15 "] in variable " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 529 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`exp(At)` := map(s ort, exponential(A, t)) : exp(A*t) = matrix(`exp(At)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#*&%\"AG\"\"\"%\"tGF)-%'matrixG6#7$ 7$,&-F%6#,$F*\"\"##F4\"\"$*&#F)F6F)-F%6#,$F*!\"%F)F),&F1#F)\"\"'*&#F)F ?F)F9F)!\"\"7$,&F1#\"\"%F6*&#FFF6F)F9F)FB,&F1F8*&F5F)F9F)F)" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 532 6 "Step 4" }{TEXT -1 44 ". Determine the exponential function of \+ \226 [" }{TEXT 530 1 "A" }{TEXT -1 16 "] in variable " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT 531 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "`exp(-As)` := map(sort, exponential(-A, s)) : exp(- A*s) = matrix(`exp(-As)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$exp G6#,$*&%\"AG\"\"\"%\"sGF*!\"\"-%'matrixG6#7$7$,&-F%6#,$F+!\"##\"\"#\" \"$*&#F*F9F*-F%6#,$F+\"\"%F*F*,&F3#F*\"\"'*&#F*FBF*F " 0 "" {MPLTEXT 1 0 74 "M(s) := evalm(`exp(-As )` &* subs(t=s, F(t))) : exp(-A*s) * F(s) = M(s) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*&%\"AG\"\"\"%\"sGF+!\"\"F+-%\"FG6#F,F +-%'matrixG6#7$7#*&,&-F&6#,$F,!\"##F+\"\"'*&#F+F=F+-F&6#,$F,\"\"%F+F-F +-F&F0F+7#*&,&F8#F+\"\"$*&#\"\"#FIF+F@F+F+F+FDF+" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "After some mathematical operations, the above matrix simplifies to the form" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "M(s) := map(combine, map(exp and, M(s), power), exp) : exp(-A*s) * F(s) = M(s) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*&%\"AG\"\"\"%\"sGF+!\"\"F+-%\"FG6#F ,F+-%'matrixG6#7$7#,&-F&6#,$F,F-#F+\"\"'*&#F+F;F+-F&6#,$F,\"\"&F+F-7#, &F7#F+\"\"$*&#\"\"#FEF+F>F+F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 537 6 "Step 6" }{TEXT -1 33 ". Compu te the definite integral " }{XPPEDIT 18 0 "Int(exp(-A*s)*F(s), s=t[0] ..t)" "6#-%$IntG6$*&-%$expG6#,$*&%\"AG\"\"\"%\"sGF-!\"\"F--%\"FG6#F.F- /F.;&%\"tG6#\"\"!F6" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "`int(M(s))` := map(sort, evalm(subs(t[0]=0, map(int, \+ M(s), s=t[0]..t)))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "In t(exp(-A*s)*F(s), s=t[0]..t) = matrix(`int(M(s))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#,$*&%\"AG\"\"\"%\"sGF.!\"\"F.-% \"FG6#F/F./F/;&%\"tG6#\"\"!F7-%'matrixG6#7$7#,(-F)6#,$F7F0#F0\"\"'*&#F .\"#IF.-F)6#,$F7\"\"&F.F0*&#F.FKF.-F)F8F.F.7#,(F@#F0\"\"$*&#\"\"#\"#:F .FHF.F.*&FMF.FNF.F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "After some mathematical operations , the above matrix simplifies to the form" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "`int(M(s))` := map(combine, map(expand, `int(M(s))` )) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int(exp(-A*s)*F(s), s=t[0]..t) = matrix(`int(M(s))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$IntG6$*&-%$expG6#,$*&%\"AG\"\"\"%\"sGF.!\"\"F.-%\"FG6#F/F./F/;&% \"tG6#\"\"!F7-%'matrixG6#7$7#,(-F)6#,$F7F0#F0\"\"'*&#F.\"#IF.-F)6#,$F7 \"\"&F.F0#F.FKF.7#,(F@#F0\"\"$*&#\"\"#\"#:F.FHF.F.FLF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 540 6 "S tep 7" }{TEXT -1 32 ". Determine the product matrix " }{XPPEDIT 18 0 "exp(A*t) * Int(exp(-A*s)*F(s),s=t[0]..t)" "6#*&-%$expG6#*&%\"AG\"\"\" %\"tGF)F)-%$IntG6$*&-F%6#,$*&F(F)%\"sGF)!\"\"F)-%\"FG6#F3F)/F3;&F*6#\" \"!F*F)" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "`exp(At) int(M(s))` := evalm(`exp(At)` &* `int(M(s))`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "exp(A*t)*Int(exp(-A*s)*F(s), s=t[0] ..t) = matrix(`exp(At) int(M(s))`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/*&-%$expG6#*&%\"AG\"\"\"%\"tGF*F*-%$IntG6$*&-F&6#,$*&F)F*%\"sGF*!\" \"F*-%\"FG6#F4F*/F4;&F+6#\"\"!F+F*-%'matrixG6#7$7#,&*&,&-F&6#,$F+\"\"# #FI\"\"$*&#F*FKF*-F&6#,$F+!\"%F*F*F*,(-F&6#,$F+F5#F5\"\"'*&#F*\"#IF*-F &6#,$F+\"\"&F*F5#F*FhnF*F*F**&,&FF#F*FW*&#F*FWF*FNF*F5F*,(FS#F5FK*&#FI \"#:F*FenF*F*FinF*F*F*7#,&*&,&FF#\"\"%FK*&#FioFKF*FNF*F5F*FRF*F**&,&FF FM*&FJF*FNF*F*F*F_oF*F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "After some mathematical operations , the above product matrix simplifies to the form" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 82 "`exp(At) int(M(s))` := map(sort, map(combine , map(expand, `exp(At) int(M(s))`))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "exp(A*t)*Int(exp(-A*s)*F(s), s=t[0]..t) = matrix(`exp (At) int(M(s))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#*&% \"AG\"\"\"%\"tGF*F*-%$IntG6$*&-F&6#,$*&F)F*%\"sGF*!\"\"F*-%\"FG6#F4F*/ F4;&F+6#\"\"!F+F*-%'matrixG6#7$7#,(-F&6#,$F+\"\"##F*\"\"'*&#F*\"#IF*-F &6#,$F+!\"%F*F**&#F*\"\"&F*-F&6#F+F*F57#,(FD#F*\"\"$*&#FG\"#:F*FMF*F5* &#F*FSF*FTF*F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 541 6 "Step 8" }{TEXT -1 26 ". Compute the sum m atrix " }{XPPEDIT 18 0 "exp(A*(t-t[0]))*C + exp(A*t)*Int(exp(-A*s)*F( s),s=t[0]..t)" "6#,&*&-%$expG6#*&%\"AG\"\"\",&%\"tGF*&F,6#\"\"!!\"\"F* F*%\"CGF*F**&-F&6#*&F)F*F,F*F*-%$IntG6$*&-F&6#,$*&F)F*%\"sGF*F0F*-%\"F G6#F>F*/F>;&F,6#F/F,F*F*" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 87 "`exp(A(t-to)) C + exp(At) int(M(s))` := evalm(`exp( A(t-to)) C` + `exp(At) int(M(s))`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "exp(A*(t-t[0])) * C + exp(A*t)*Int(exp(-A*s)*F(s), s =t[0]..t) = matrix(`exp(A(t-to)) C + exp(At) int(M(s))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$expG6#*&%\"AG\"\"\",&%\"tGF+&F-6#\" \"!!\"\"F+F+%\"CGF+F+*&-F'6#*&F*F+F-F+F+-%$IntG6$*&-F'6#,$*&F*F+%\"sGF +F1F+-%\"FG6#F?F+/F?;F.F-F+F+-%'matrixG6#7$7#,(-F'6#,$F-!\"%#\"#J\"#I* &#F+\"\"'F+-F'6#,$F-\"\"#F+F+*&#F+\"\"&F+-F'6#F-F+F17#,(FK#!#i\"#:*&#F +\"\"$F+FUF+F+*&#F+FenF+FfnF+F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 546 6 "Step 9" }{TEXT -1 30 ". Obta in the solution matrix [" }{TEXT 545 1 "X" }{TEXT 543 1 "(" }{TEXT 542 1 "t" }{TEXT 544 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 71 "X(t) := evalm(`exp(A(t-to)) C + exp(At) int(M(s))`) : 'X(t)' = X(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6#%\"tG -%'matrixG6#7$7#,(-%$expG6#,$F'!\"%#\"#J\"#I*&#\"\"\"\"\"'F8-F/6#,$F' \"\"#F8F8*&#F8\"\"&F8-F/F&F8!\"\"7#,(F.#!#i\"#:*&#F8\"\"$F8F:F8F8*&#F8 F@F8FAF8FB" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 729 21 "Clearing of fractions" }{TEXT -1 273 " may be a pplied to the elements of this matrix by multiplying them by the least common multiple of denominators of their numeric operands and dividin g the new matrix by the same. If this to be done manually, it is intui tive and simple. If the task is to be performed with " }{TEXT 728 5 "M aple" }{TEXT -1 297 ", the procedure is quite elaborate and requires s everal operations. A suitable method proposed below has been developed with a view to providing a universal approach to tackling more compli cated cases. The method also works when some of the numeric operands o f matrix elements are natural numbers." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "(a) Create a list of the elements of [" }{TEXT 733 1 "X" }{TEXT 731 1 "(" }{TEXT 730 1 "t" }{TEXT 732 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "L := convert(convert(X(t), set), list) : elements(' X(t)') = L ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)elementsG6#-%\"XG6 #%\"tG7$,(-%$expG6#,$F*!\"%#\"#J\"#I*&#\"\"\"\"\"'F7-F.6#,$F*\"\"#F7F7 *&#F7\"\"&F7-F.F)F7!\"\",(F-#!#i\"#:*&#F7\"\"$F7F9F7F7*&#F7F?F7F@F7FA " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "(b) Extract denominators of the numeric operands of the e lements of the list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "fo r i to nops(L) do for n to nops(op(i, L)) do den(el[i,n]('L')) := de nom(op(n, L[i])) : print(evaln(denominator(element[i,n]('L'))) = den (el[i,n]('L'))) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% ,denominatorG6#-&%(elementG6$\"\"\"F+6#%\"LG\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6$\"\"\"\"\"#6#%\"LG\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6$\" \"\"\"\"$6#%\"LG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominat orG6#-&%(elementG6$\"\"#\"\"\"6#%\"LG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6$\"\"#F+6#%\"LG\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6$\"\"# \"\"$6#%\"LG\"\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "(c) Rename the denominators by indexing e ach with a single-digit subscript:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "k := 0 : for i to nops(L) do for n to nops(op(i, \+ L)) do k := k+1 : den(el[k]('L')) := den(el[i,n]('L')) : print(ev aln(denominator(element[k]('L'))) = den(el[k]('L'))) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6#\"\" \"6#%\"LG\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&% (elementG6#\"\"#6#%\"LG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,de nominatorG6#-&%(elementG6#\"\"$6#%\"LG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6#\"\"%6#%\"LG\"#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(elementG6#\"\"&6 #%\"LG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,denominatorG6#-&%(e lementG6#\"\"'6#%\"LG\"\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "(d) Count the total number of den ominators of the numeric operands of the elements of the list:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "num_ops('L') := k : 'num_o ps(L)' = num_ops('L') ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(num_ops G6#%\"LG\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "(e) Create a sequence of denominators of \+ the numeric operands of the elements of the list:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 156 "`seq(den(el(L)))` := [seq(den(el[m]('L')), \+ m=1..num_ops('L'))] : den(el('L')) := op(`seq(den(el(L)))`) : deno minators(elements('X(t)')) = den(el('L')) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-denominatorsG6#-%)elementsG6#-%\"XG6#%\"tG6(\"#I\" \"'\"\"&\"#:\"\"$F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "(f) Find the least common multiple of denominators of the numeric operands of the elements of [" }{TEXT 737 1 "X" }{TEXT 735 1 "(" }{TEXT 734 1 "t" }{TEXT 736 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "LCM := lcm(den( el('L'))) : least_common_multiple(denominators('X(t)')) = LCM ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%6least_common_multipleG6#-%-denomin atorsG6#-%\"XG6#%\"tG\"#I" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "(g) Multiply the elements of [" } {TEXT 741 1 "X" }{TEXT 739 1 "(" }{TEXT 738 1 "t" }{TEXT 740 1 ")" } {TEXT -1 111 "] by the least common multiple of denominators of their \+ numeric operands and divide the new matrix by the same:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "X(t) := evalm(X(t) * LCM)/LCM : ' X(t)' = X(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6#%\"tG,$-%'m atrixG6#7$7#,(-%$expG6#,$F'!\"%\"#J*&\"\"&\"\"\"-F06#,$F'\"\"#F7F7*&\" \"'F7-F0F&F7!\"\"7#,(F/!$C\"*&\"#5F7F8F7F7*&F=F7F>F7F?#F7\"#I" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 547 7 "Step 10" }{TEXT -1 22 ". Verify the solution:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(a) com pute the derivative " }{XPPEDIT 18 0 "Diff(X(t),t)" "6#-%%DiffG6$-%\" XG6#%\"tGF)" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "X(t) := evalm(X(t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`der(X)` := map(diff, X(t), t) : Diff('X(t)', t) = matrix(`der(X )`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"XG6#%\"tGF*-%' matrixG6#7$7#,(-%$expG6#,$F*!\"%#!#i\"#:*&#\"\"\"\"\"$F;-F26#,$F*\"\"# F;F;*&#F;\"\"&F;-F2F)F;!\"\"7#,(F1#\"$[#F8*&#F@F " 0 "" {MPLTEXT 1 0 68 "`der(X)` := evalm(`der(X)` * 15)/15 : Diff('X(t)', t) = `der(X)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%%DiffG6$-%\"XG6#%\"tGF*,$-%'matrixG6#7$7#,(-%$expG6#,$F*!\"%!#i*&\" \"&\"\"\"-F36#,$F*\"\"#F:F:*&\"\"$F:-F3F)F:!\"\"7#,(F2\"$[#*&\"#5F:F;F :F:*&F@F:FAF:FB#F:\"#:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "(b) compute the sum [" }{TEXT 551 1 "A" }{TEXT -1 2 "][" }{TEXT 552 1 "X" }{TEXT 549 1 "(" }{TEXT 548 1 "t" }{TEXT 550 1 ")" }{TEXT -1 1 "]" }{TEXT 652 3 " + " }{TEXT -1 1 "[" }{TEXT 556 1 "F" }{TEXT 553 1 "(" }{TEXT 554 1 "t" }{TEXT 555 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "`AX + F` := evalm(A &* X(t) + F(t)) : A * 'X(t)' + 'F(t)' = matr ix(`AX + F`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"AG\"\"\"-%\" XG6#%\"tGF'F'-%\"FGF*F'-%'matrixG6#7$7#,(-%$expG6#,$F+!\"%#!#i\"#:*&#F '\"\"$F'-F56#,$F+\"\"#F'F'*&#F'\"\"&F'-F5F*F'!\"\"7#,(F4#\"$[#F;*&#FBF >F'F?F'F'*&#F'FEF'FFF'FG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "or, after manually clearing the fr actions," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "`AX + F` := eva lm(`AX + F` * 15)/15 : A * 'X(t)' + 'F(t)' = `AX + F` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"AG\"\"\"-%\"XG6#%\"tGF'F'-%\"FGF*F',$-% 'matrixG6#7$7#,(-%$expG6#,$F+!\"%!#i*&\"\"&F'-F66#,$F+\"\"#F'F'*&\"\"$ F'-F6F*F'!\"\"7#,(F5\"$[#*&\"#5F'F=F'F'*&FBF'FCF'FD#F'\"#:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 722 5 "equal" }{TEXT -1 64 " function applied to the result ant matrices of (a) and (b), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "equal(`der(X)`, `AX + F`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "returns the " }{TEXT 723 7 "Boole an" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 75 ", \+ which verifies that both matrices are equal. This verifies the solutio n." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 285 "" 0 " " {TEXT 717 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 272 "" 0 "" {TEXT 658 67 "------------------------------- ------------------------------------" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }