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-1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 259 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 259 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 259 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 491 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 493 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 494 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(27)" }{TEXT 313 1 " " }{TEXT 312 48 "Differentiation o f matrices comprising functions" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 517 10 "OBJECTIVES" }{TEXT 518 1 ": " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 519 1 "\225" }{TEXT -1 96 " To state the condition necessary fo r a matrix containing functions of one variable to have a " }{TEXT 520 10 "derivative" }{TEXT -1 32 " with respect to this variable." }} }{EXCHG {PARA 0 "" 0 "" {TEXT 533 1 "\225" }{TEXT -1 33 " To introduc e the concept of a " }{TEXT 534 14 "differentiable" }{TEXT -1 9 " ma trix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 529 1 "\225" }{TEXT -1 84 " To provide a definition of matrix derivative in the form of a symbolic e xpression." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 530 1 "\225" }{TEXT -1 93 " To provide an example of a derivative of a rectangular matrix whose elements are functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 540 1 "\225 " }{TEXT -1 127 " To provide an example how to determine the largest \+ interval about the origin, in which the derivative of a matrix is defi ned." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 531 1 "\225" }{TEXT -1 65 " To \+ specify and illustrate properties of matrix differentiation." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "restart : with(linalg, equal, inverse, orthog, tran spose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 37 "If some or all elements of a matrix [" }{TEXT 516 1 "A" }{TEXT -1 7 "] are " }{TEXT 509 9 "functions" }{TEXT -1 5 " of " }{TEXT 510 3 "one" }{TEXT -1 11 " variable " }{TEXT 511 1 "t" } {TEXT 512 1 "," }{TEXT -1 44 " it is convenient to denote the matrix \+ as [" }{TEXT 513 1 "A" }{TEXT 515 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 514 1 ")" }{TEXT -1 52 "] when considering differentiation of su ch a matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "When the elements of a matrix [" }{TEXT 258 1 " A" }{TEXT 315 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 314 1 ")" } {TEXT -1 22 "] are functions of a " }{TEXT 497 4 "real" }{TEXT -1 12 " variable " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 7 " and " } {TEXT 259 3 "all" }{TEXT -1 38 " are differentiable with respect to \+ " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 27 " in some common interval " }{XPPEDIT 18 0 "t[0]" "6#&%\"tG6#\"\"!" }{TEXT -1 1 " " }{TEXT 521 1 "<" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT 522 1 "," }{TEXT -1 10 " then a " }{TEXT 261 10 "derivative" }{TEXT -1 6 " of [" } {TEXT 260 1 "A" }{TEXT 317 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 316 1 ")" }{TEXT -1 19 "] with respect to " }{XPPEDIT 18 0 "t" "6#%\" tG" }{TEXT -1 41 " may be defined. Then, also the matrix [" }{TEXT 523 1 "A" }{TEXT 525 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 524 1 " )" }{TEXT -1 17 "] is said to be " }{TEXT 528 14 "differentiable" } {TEXT -1 19 " in the interval " }{XPPEDIT 18 0 "t[0]" "6#&%\"tG6#\" \"!" }{TEXT -1 1 " " }{TEXT 526 1 "<" }{TEXT -1 1 " " }{XPPEDIT 18 0 " t" "6#%\"tG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\" " }{TEXT 527 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 400 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 262 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 401 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 263 1 "A" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 11 ")] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A(t) := matrix(2, 3, [a[11] (t), a[12](t), a[13](t), a[21](t), a[22](t), a[23](t)]) : 'A(t)' = A (t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7 %-&%\"aG6#\"#6F&-&F/6#\"#7F&-&F/6#\"#8F&7%-&F/6#\"#@F&-&F/6#\"#AF&-&F/ 6#\"#BF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 19 "The derivative of [" }{TEXT 264 1 "A" }{TEXT 319 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 318 1 ")" }{TEXT -1 19 "] wi th respect to " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 63 " is defin ed to be the matrix of the same order with elements " }{XPPEDIT 18 0 "Diff(a[ij],t)" "6#-%%DiffG6$&%\"aG6#%#ijG%\"tG" }{TEXT -1 8 " , viz. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`der(A)` := map(Diff, A(t), t) : Diff('A(t)', t) = \+ matrix(`der(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"A G6#%\"tGF*-%'matrixG6#7$7%-F%6$-&%\"aG6#\"#6F)F*-F%6$-&F46#\"#7F)F*-F% 6$-&F46#\"#8F)F*7%-F%6$-&F46#\"#@F)F*-F%6$-&F46#\"#AF)F*-F%6$-&F46#\"# BF)F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation can be displayed in \"like-in-a-book\" \+ form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Diff(A(t), t) = matrix(`der(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-% 'matrixG6#7$7%-&%\"aG6#\"#66#%\"tG-&F.6#\"#7F1-&F.6#\"#8F17%-&F.6#\"#@ F1-&F.6#\"#AF1-&F.6#\"#BF1F2-F(6#7$7%-F%6$F,F2-F%6$F3F2-F%6$F7F27%-F%6 $F " 0 "" {MPLTEXT 1 0 98 "A(t) := matrix(2, 3, [cosh(t), sin( t), cosh(2*t), sinh(t), cos(t), sinh(2*t)]) : 'A(t)' = A(t) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%-%%coshG F&-%$sinGF&-F.6#,$F'\"\"#7%-%%sinhGF&-%$cosGF&-F7F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The der ivative of [" }{TEXT 267 1 "A" }{TEXT 323 1 "(" }{XPPEDIT 18 0 "t" "6# %\"tG" }{TEXT 322 1 ")" }{TEXT -1 26 "] is the following matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`der(A)` := map(diff, A(t), \+ t) : Diff('A(t)', t) = matrix(`der(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"AG6#%\"tGF*-%'matrixG6#7$7%-%%sinhGF)-%$ cosGF),$-F16#,$F*\"\"#F87%-%%coshGF),$-%$sinGF)!\"\",$-F;F6F8" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Diff(A(t), t) = matrix(`der(A)`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%DiffG6$-%'matrixG6#7$7%-%%coshG6#%\"tG-%$sinGF.-F -6#,$F/\"\"#7%-%%sinhGF.-%$cosGF.-F8F3F/-F(6#7$7%F7F9,$F;F57%F,,$F0!\" \",$F2F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "The following example illustrates how to determine the " }{TEXT 536 7 "largest" }{TEXT -1 68 " interval about the origin , in which a matrix derivative is defined." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Find the largest \+ interval about the origin, in which the derivative of a matrix [" } {TEXT 537 1 "A" }{TEXT -1 24 "] is defined, given that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "A(t) := matrix(2, 3, [2*t^4, tan(t) , ln(t^2-3)/2, 2*cos(t), t^2+6, cosh(t)]) : 'A(t)' = A(t) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%,$*$)F' \"\"%\"\"\"\"\"#-%$tanGF&,$-%#lnG6#,&*$)F'F2F1F1\"\"$!\"\"#F1F27%,$-%$ cosGF&F2,&F:F1\"\"'F1-%%coshGF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 539 6 "Step 1" }{TEXT -1 29 ". Comp ute the derivative of [" }{TEXT 538 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`der(A)` := map(diff, A(t), t) : \+ Diff('A(t)', t) = matrix(`der(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%DiffG6$-%\"AG6#%\"tGF*-%'matrixG6#7$7%,$*$)F*\"\"$\"\"\"\"\"),&F 4F4*$)-%$tanGF)\"\"#F4F4*&F*F4,&*$)F*F;F4F4F3!\"\"F@7%,$-%$sinGF)!\"#, $F*F;-%%sinhGF)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "It may be easily noticed that the functio ns at locations " }{TEXT 543 5 "(1,2)" }{TEXT -1 7 " and " }{TEXT 544 5 "(1,3)" }{TEXT -1 121 " of this matrix have discontinuities. Th is implies that the matrix is not defined over the entire field of rea l numbers." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Consequently, it is necessary to determine the i nterval of either function in which it is continuous and find its part common to both functions." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 545 6 "Step 2" }{TEXT -1 75 ". Find the \+ interval about the origin within which the element at location " } {TEXT 542 5 "(1,2)" }{TEXT -1 16 " is continuous:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "It is a w ell-known fact that the branch of the function " }{XPPEDIT 18 0 "y=ta n(t)" "6#/%\"yG-%$tanG6#%\"tG" }{TEXT -1 84 " that goes through the o rigin is asymptotic to the lines described by the equations" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "t[n_a] := -Pi/2 : t[p_a] : = Pi/2 : 't[n_a]' = t[n_a] ; 't[p_a]' = t[p_a] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#%$n_aG,$%#PiG#!\"\"\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"tG6#%$p_aG,$%#PiG#\"\"\"\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where \+ " }{XPPEDIT 18 0 "t[n_a]" "6#&%\"tG6#%$n_aG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "t[p_a]" "6#&%\"tG6#%$p_aG" }{TEXT -1 64 " are the end \+ points of the interval with the central point at " }{XPPEDIT 18 0 "t= 0" "6#/%\"tG\"\"!" }{TEXT -1 42 " within which the function is conti nuous." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Consequently, the branch of the function " } {XPPEDIT 18 0 "y=1+tan(t)^2" "6#/%\"yG,&\"\"\"F&*$-%$tanG6#%\"tG\"\"#F &" }{TEXT -1 75 " is also asymptotic to the same lines and its values at both end points, " }{XPPEDIT 18 0 "t[n_a]" "6#&%\"tG6#%$n_aG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "t[p_a]" "6#&%\"tG6#%$p_aG" } {TEXT 546 1 "," }{TEXT -1 27 " tend to infinity as shown" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "Limit([`der(A)`[1,2]], t=t[n_a]) = limit(`der(A)`[1,2], t=t[n_a]) ;\nLimit([`der(A)`[1,2]], t=t[p_a]) = \+ limit(`der(A)`[1,2], t=t[p_a]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %&LimitG6$7#,&\"\"\"F)*$)-%$tanG6#%\"tG\"\"#F)F)/F/,$%#PiG#!\"\"F0%)in finityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$7#,&\"\"\"F)*$) -%$tanG6#%\"tG\"\"#F)F)/F/,$%#PiG#F)F0%)infinityG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Therefore, the branch of the function " }{XPPEDIT 18 0 "y=1+tan(t)^2" "6#/%\"yG ,&\"\"\"F&*$-%$tanG6#%\"tG\"\"#F&" }{TEXT -1 67 " that goes through t he origin is continuous in the open interval " }{TEXT 547 1 "(" } {XPPEDIT 18 0 "-Pi" "6#,$%#PiG!\"\"" }{TEXT 548 4 "/2, " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 549 4 "/2)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 551 6 "Step 3" }{TEXT -1 75 ". Find the interval about the origin within which the element at locati on " }{TEXT 550 5 "(1,3)" }{TEXT -1 16 " is continuous:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "I t may be easily noticed that this function is discontinuous if its den ominator becomes " }{XPPMATH 20 "6#%%zeroG" }{TEXT 541 1 "." }{TEXT -1 31 " Equating the denominator to " }{XPPMATH 20 "6#%%zeroG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Eq_denom(el13) := denom(`der (A)`[1,3]) = 0 : Eq_denom(el13) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,&*$)%\"tG\"\"#\"\"\"F)\"\"$!\"\"\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "and solving the r esultant equation yield" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " solution := solve(\{Eq_denom(el13)\}) : solution ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#/%\"tG*$-%%sqrtG6#\"\"$\"\"\"<#/F%,$F&!\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Let the numerical parts of the solution be assigned the names \+ " }{XPPEDIT 18 0 "t[n_d]" "6#&%\"tG6#%$n_dG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "t[p_d]" "6#&%\"tG6#%$p_dG" }{TEXT 552 1 "." }{TEXT -1 42 " Thus, the points at which the function " }{XPPEDIT 18 0 "y=t" " 6#/%\"yG%\"tG" }{TEXT -1 1 "/" }{TEXT 553 1 "(" }{XPPEDIT 18 0 "t^2-3 " "6#,&*$%\"tG\"\"#\"\"\"\"\"$!\"\"" }{TEXT 554 1 ")" }{TEXT -1 22 " \+ is discontinuous are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "t[ n_d] := rhs(solution[2][1]) : t[p_d] := rhs(solution[1][1]) : 't[n _d]' = t[n_d] ; 't[p_d]' = t[p_d] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#%$n_dG,$*$-%%sqrtG6#\"\"$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#%$p_dG*$-%%sqrtG6#\"\"$\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "T he points " }{XPPEDIT 18 0 "t[n_a]" "6#&%\"tG6#%$n_aG" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "t[p_a]" "6#&%\"tG6#%$p_aG" }{TEXT -1 64 " are the end points of the interval with the central point at " } {XPPEDIT 18 0 "t= 0" "6#/%\"tG\"\"!" }{TEXT -1 53 " within which the \+ function exists and is continuous." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The function " } {XPPEDIT 18 0 "y=t" "6#/%\"yG%\"tG" }{TEXT -1 1 "/" }{TEXT 555 1 "(" } {XPPEDIT 18 0 "t^2-3" "6#,&*$%\"tG\"\"#\"\"\"\"\"$!\"\"" }{TEXT 556 1 ")" }{TEXT -1 106 " is not only discontinuous at both end points but \+ also it has no limiting values at these points as shown" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "Limit([`der(A)`[1,3]], t=t[n_d]) = limit(`der(A)`[1,3], t=t[n_d]) ;\nLimit([`der(A)`[1,3]], t=t[p_d]) = \+ limit(`der(A)`[1,3], t=t[p_d]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %&LimitG6$7#*&%\"tG\"\"\",&*$)F)\"\"#F*F*\"\"$!\"\"F0/F),$*$-%%sqrtG6# F/F*F0%*undefinedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$7#*& %\"tG\"\"\",&*$)F)\"\"#F*F*\"\"$!\"\"F0/F)*$-%%sqrtG6#F/F*%*undefinedG " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Therefore, the function " }{XPPEDIT 18 0 "y=t" "6#/%\"yG %\"tG" }{TEXT -1 1 "/" }{TEXT 557 1 "(" }{XPPEDIT 18 0 "t^2-3" "6#,&*$ %\"tG\"\"#\"\"\"\"\"$!\"\"" }{TEXT 558 1 ")" }{TEXT -1 38 " is contin uous in the open interval " }{TEXT 559 1 "(" }{XPPEDIT 18 0 "-sqrt(3) " "6#,$-%%sqrtG6#\"\"$!\"\"" }{TEXT 560 2 ", " }{XPPEDIT 18 0 "sqrt(3) " "6#-%%sqrtG6#\"\"$" }{TEXT 561 2 ")." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 562 6 "Step 4" }{TEXT -1 50 ". Determine the interval common to both functions:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Si nce" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t[n_d] < t[n_a] ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#2,$*$-%%sqrtG6#\"\"$\"\"\"!\"\",$%#Pi G#F+\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t [p_a] < t[p_d] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2,$%#PiG#\"\"\"\" \"#*$-%%sqrtG6#\"\"$F'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "both functions are continuous with in the interval " }{XPPEDIT 18 0 "-Pi" "6#,$%#PiG!\"\"" }{TEXT -1 3 " /2 " }{TEXT 564 2 "< " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 1 " " } {TEXT 565 1 "<" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 34 "/2 and the derivative of matrix [" }{TEXT 563 1 "A" }{TEXT -1 30 "] is defined in this interval." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 535 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 4 "N .B." }{TEXT -1 20 " If two matrices, [" }{TEXT 268 1 "A" }{TEXT 325 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 324 1 ")" }{TEXT -1 7 "] and [" }{TEXT 269 1 "B" }{TEXT 327 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 326 1 ")" }{TEXT -1 159 "], are conformable for addition and dif ferentiable in some common interval, the derivative of their sum is eq ual to the sum of the derivatives of both matrices" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff([A(t)+B(t)],t)=Diff(A(t),t)+Diff(B(t), t)" "6#/-%%DiffG6$7#,&-%\"AG6#%\"tG\"\"\"-%\"BG6#F,F-F,,&-F%6$-F*6#F,F ,F--F%6$-F/6#F,F,F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "For example, let " }{TEXT 404 1 " (" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 405 3 " \327 " }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 406 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 271 1 "A" }{TEXT 329 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 328 1 ")" } {TEXT -1 7 "] and [" }{TEXT 272 1 "B" }{TEXT 331 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 330 1 ")" }{TEXT -1 13 "] be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "A(t) := matrix(2, 3, [t, t^3, -1, \+ t^2, 2*t, -3*t^2]) : B(t) := matrix(2, 3, [2, t^2, t^3, -2*t, 3*t^2, -1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "'A(t)' = A(t) ; 'B(t)' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%' matrixG6#7$7%F'*$)F'\"\"$\"\"\"!\"\"7%*$)F'\"\"#F0,$F'F5,$F3!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'matrixG6#7$7%\"\"#*$) F'F-\"\"\"*$)F'\"\"$F07%,$F'!\"#,$F.F3!\"\"" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "(a) The derivat ive of the sum of the two matrices is the following " }{TEXT 407 1 "( " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 408 3 " \327 " }{XPPEDIT 18 0 "3 " "6#\"\"$" }{TEXT 409 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`der(A+B)` := map(factor, map(diff, evalm (A(t)+B(t)), t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Diff( ['A(t)' + 'B(t)'], t) = matrix(`der(A+B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#,&-%\"AG6#%\"tG\"\"\"-%\"BGF+F-F,-%'matrix G6#7$7%F-*&F,F-,&\"\"#F-*&\"\"$F-F,F-F-F-,$*$)F,F7F-F97%,&F,F7F7!\"\", &F7F-*&\"\"'F-F,F-F-,$F,!\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "(b) The sum of the derivatives \+ of both matrices is the following " }{TEXT 410 1 "(" }{XPPEDIT 18 0 " 2" "6#\"\"#" }{TEXT 411 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 412 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "`der(A) + der(B)` := map(factor, evalm(map(diff, A(t) , t) + map(diff, B(t), t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Diff('A(t)', t) + Diff('B(t)', t) = matrix(`der(A) + der(B)`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%DiffG6$-%\"AG6#%\"tGF+\"\"\" -F&6$-%\"BGF*F+F,-%'matrixG6#7$7%F,*&F+F,,&\"\"#F,*&\"\"$F,F+F,F,F,,$* $)F+F8F,F:7%,&F+F8F8!\"\",&F8F,*&\"\"'F,F+F,F,,$F+!\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "B oth matrices of (a) and (b) are equal by inspection." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 273 5 "* * * " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 4 "N.B." }{TEXT -1 129 " The derivative of the product of a scalar (number) and a matrix is equal to the product of the scalar an d the matrix derivative" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Di ff([k*A(t)], t) = k*Diff(A(t), t)" "6#/-%%DiffG6$7#*&%\"kG\"\"\"-%\"AG 6#%\"tGF*F.*&F)F*-F%6$-F,6#F.F.F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a \+ " }{TEXT 413 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 414 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 415 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 275 1 "A" }{TEXT 333 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 332 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A(t) := matrix(2, 3, [t, t^3, -1, t^2, 2*t, -3*t^2]) \+ : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG- %'matrixG6#7$7%F'*$)F'\"\"$\"\"\"!\"\"7%*$)F'\"\"#F0,$F'F5,$F3!\"$" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and the scalar " }{XPPEDIT 18 0 "k=3" "6#/%\"kG\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "k := 3 :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "(a) The derivative of the pr oduct of the scalar and the matrix is the following " }{TEXT 416 1 "( " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 417 3 " \327 " }{XPPEDIT 18 0 "3 " "6#\"\"$" }{TEXT 418 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`der(kA)` := map(diff, evalm(k * A(t)), t ) : Diff(['k*A(t)'], t) = matrix(`der(kA)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*&%\"kG\"\"\"-%\"AG6#%\"tGF*F.-%'matrixG6# 7$7%\"\"$,$*$)F.\"\"#F*\"\"*\"\"!7%,$F.\"\"'F=,$F.!#=" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "( b) The product of the scalar and the matrix derivative is the followin g " }{TEXT 419 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 420 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 421 1 ")" }{TEXT -1 9 " matrix :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`k der(A)` := evalm(k \+ * map(diff, A(t), t)) : 'k' * Diff('A(t)', t) = matrix(`k der(A)`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"kG\"\"\"-%%DiffG6$-%\"AG6#% \"tGF-F&-%'matrixG6#7$7%\"\"$,$*$)F-\"\"#F&\"\"*\"\"!7%,$F-\"\"'F<,$F- !#=" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 53 "Both matrices of (a) and (b) are equal by inspection." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 276 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 4 "N.B." }{TEXT -1 20 " If two matrices, [ " }{TEXT 278 1 "A" }{TEXT 335 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 334 1 ")" }{TEXT -1 7 "] and [" }{TEXT 279 1 "B" }{TEXT 337 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 336 1 ")" }{TEXT -1 282 "], are \+ conformable for multiplication and differentiable in some common inter val, the derivative of their product is equal to the product of the de rivative of the pre-multiplier and the post-multiplier plus the produc t of the pre-multiplier and the derivative of the post-multiplier" }}} {EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff([A(t)*B(t)],t) = Diff(A(t ),t) * B(t) + A(t) * Diff(B(t),t)" "6#/-%%DiffG6$7#*&-%\"AG6#%\"tG\"\" \"-%\"BG6#F,F-F,,&*&-F%6$-F*6#F,F,F--F/6#F,F-F-*&-F*6#F,F--F%6$-F/6#F, F,F-F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 422 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 423 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 424 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 280 1 "A" } {TEXT 339 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 338 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A(t) : = matrix(2, 3, [t, t^3, -1, t^2, 2*t, -3*t^2]) : 'A(t)' = A(t) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%F'*$)F' \"\"$\"\"\"!\"\"7%*$)F'\"\"#F0,$F'F5,$F3!\"$" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "and a " }{TEXT 425 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 426 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 427 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 281 1 "B" }{TEXT 341 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 340 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "B(t) := matrix(3, 3, [2*t^2, -3*t, t, t^3, -3*t^2, 2, 2*t^3, -2*t, 2*t^2]) : 'B(t)' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'matrixG6#7%7%,$*$)F'\"\"#\"\"\"F0,$F'! \"$F'7%*$)F'\"\"$F1,$F.F3F07%,$F5F0,$F'!\"#F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "(a) The deriva tive of the matrix product is the following " }{TEXT 428 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 429 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 430 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "`der(AB)` := map(sort, map(factor, map(diff, \+ evalm(A(t) &* B(t)), t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Diff(['A(t)' * 'B(t)'], t) = matrix(`der(AB)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*&-%\"AG6#%\"tG\"\"\"-%\"BGF+F-F,-%'mat rixG6#7$7%,$*$)F,\"\"&F-\"\"',(*$)F,\"\"%F-!#:*&F9F-F,F-!\"\"\"\"#F-,$ *&,&F,\"\"$F-F@F-F,F-FA7%,$*&,&F,\"#:\"\")F@F-)F,FEF-!\"#,$*$)F,FAF-! \"*,(*$FLF-!#C*&FEF-FPF-F-F=F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "(b) The product of the deriv ative of the pre-multiplier and the post-multiplier plus the product o f the pre-multiplier and the derivative of the post-multiplier is the \+ following " }{TEXT 431 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 432 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 433 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "`der(A) B + A der(B)` := map(sort, map(factor, evalm(map(diff, A(t), t) &* B(t) + A (t) &* map(diff, B(t), t)))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Diff('A(t)', t) * 'B(t)' + 'A(t)' * Diff('B(t)', t) = matrix(` der(A) B + A der(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%%Dif fG6$-%\"AG6#%\"tGF,\"\"\"-%\"BGF+F-F-*&F)F--F'6$F.F,F-F--%'matrixG6#7$ 7%,$*$)F,\"\"&F-\"\"',(*$)F,\"\"%F-!#:*&F " 0 "" {MPLTEXT 1 0 41 "equal(`der(AB)`, `der( A) B + A der(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "returns the " }{TEXT 566 7 "Boolean" }{TEXT -1 9 " value " } {XPPMATH 20 "6#%%trueG" }{TEXT -1 47 ", which verifies that both matr ices are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 305 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 306 4 "N.B." }{TEXT -1 55 " I t follows immediately from the above ( by setting [" }{TEXT 307 1 "B " }{TEXT 343 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 342 1 ")" } {TEXT -1 1 "]" }{TEXT 434 3 " = " }{TEXT -1 1 "[" }{TEXT 308 1 "A" } {TEXT 345 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 344 1 ")" }{TEXT -1 31 "] ) that for a square matrix [" }{TEXT 309 1 "A" }{TEXT 347 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 346 1 ")" }{TEXT -1 14 "], in \+ general," }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff([A(t)]^2,t) " "6#-%%DiffG6$*$7#-%\"AG6#%\"tG\"\"#F+" }{TEXT -1 3 " " }{TEXT 19 15 "is not equal to" }{TEXT -1 3 " " }{XPPEDIT 18 0 "2*A(t)*Diff(A(t ),t)" "6#*(\"\"#\"\"\"-%\"AG6#%\"tGF%-%%DiffG6$-F'6#F)F)F%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "bu t" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff([A(t)]^2,t) = Diff( A(t),t)*A(t) + A(t)*Diff(A(t),t)" "6#/-%%DiffG6$*$7#-%\"AG6#%\"tG\"\"# F,,&*&-F%6$-F*6#F,F,\"\"\"-F*6#F,F4F4*&-F*6#F,F4-F%6$-F*6#F,F,F4F4" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 435 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 436 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 437 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 310 1 "A" }{TEXT 349 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 348 1 ")" }{TEXT -1 10 "] given as " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "A(t) := matrix(2, 2, [2 , t^2, t^3, 1]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"AG6#%\"tG-%'matrixG6#7$7$\"\"#*$)F'F-\"\"\"7$*$)F'\"\"$F0F0" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "(a) The derivative of the matrix squared is the following " } {TEXT 438 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 439 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 440 1 ")" }{TEXT -1 9 " matrix:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`der(A^2)` := map(diff, ev alm(A(t)^2), t) : Diff(['A(t)']^2, t) = matrix(`der(A^2)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*$)7#-%\"AG6#%\"tG\"\"#\"\" \"F--%'matrixG6#7$7$,$*$)F-\"\"%F/\"\"&,$F-\"\"'7$,$*$)F-F.F/\"\"*F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "(b) The doubled product of the matrix and its derivative \+ is the following " }{TEXT 441 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 442 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 443 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "` 2A der(A)` := evalm(2*A(t) &* map(diff, A(t), t)) : 2 * 'A(t)' * Dif f('A(t)', t) = matrix(`2A der(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/,$*&-%\"AG6#%\"tG\"\"\"-%%DiffG6$F&F)F*\"\"#-%'matrixG6#7$7$,$*$)F) \"\"%F*\"\"',$F)\"\")7$,$*$)F)F.F*F8,$F5F7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "which is evidentl y " }{TEXT 311 3 "not" }{TEXT -1 47 " equal to the derivative of the m atrix squared." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "According to the formula for the derivati ve of [" }{TEXT 498 1 "A" }{TEXT 500 1 "(" }{XPPEDIT 18 0 "t" "6#%\"t G" }{TEXT 499 1 ")" }{TEXT -1 1 "]" }{TEXT 501 3 "^2," }{TEXT -1 120 " its right-hand side is needed for display in the next step. This, ho wever, turns out to be normally impossible due to " }{TEXT 503 5 "Mapl e" }{TEXT 502 1 "\222" }{TEXT -1 56 "s internal evaluations, which yie ld this unwanted result" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " Diff('A(t)', t) * 'A(t)' + 'A(t)' * Diff('A(t)', t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%\"AG6#%\"tG\"\"\"-%%DiffG6$F%F(F)\"\"#" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To get around this problem, a special undocumented command " } {TEXT 569 6 "gensym" }{TEXT -1 28 " must be used. This command " } {TEXT 567 3 "gen" }{TEXT -1 9 "erates a " }{TEXT 568 3 "syn" }{TEXT -1 140 "onym for a variable, i.e. a variable, which looks the same as \+ the original one, but has a different machine address and so is consid ered by " }{TEXT 570 5 "Maple" }{TEXT -1 17 " to be different." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "(c) The product of the matrix derivative and the matrix plus t he product of the matrix and its derivative is the following " } {TEXT 444 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 445 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 446 1 ")" }{TEXT -1 9 " matrix:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "`der(A) A + A der(A)` := \+ evalm(evalm(map(diff, A(t), t) &* A(t)) + evalm(A(t) &* map(diff, A(t) , t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "Diff('A(t)', t) * `tools/gensym`(A)(t) + 'A(t)' * Diff('A(t)', t) = matrix(`der(A) A \+ + A der(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%%DiffG6$-%\"A G6#%\"tGF,\"\"\"-F*F+F-F-*&F)F-F&F-F--%'matrixG6#7$7$,$*$)F,\"\"%F-\" \"&,$F,\"\"'7$,$*$)F,\"\"#F-\"\"*F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "which is equal to the d erivative of the matrix squared." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 366 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 367 4 "N.B." } {TEXT -1 15 " If a matrix [" }{TEXT 368 1 "A" }{TEXT 370 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 369 1 ")" }{TEXT -1 9 "] is an " } {TEXT 371 10 "orthogonal" }{TEXT -1 19 " matrix, then and " }{TEXT 372 4 "only" }{TEXT -1 5 " then" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff(A(t)^2,t) = 2*A(t)*Diff(A(t),t)" "6#/-%%DiffG6$*$-%\"AG6#% \"tG\"\"#F+*(F,\"\"\"-F)6#F+F.-F%6$-F)6#F+F+F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 447 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 448 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 449 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 373 1 "A" }{TEXT 375 1 "(" }{XPPEDIT 18 0 "t " "6#%\"tG" }{TEXT 374 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A(t) := matrix(2, 2, [cos(t), -sin( t), sin(t), cos(t)]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7$-%$cosGF&,$-%$sinGF&!\"\"7$F0F-" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(a) Check whether [" }{TEXT 397 1 "A" }{TEXT 399 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 398 1 ")" }{TEXT -1 26 "] is an ort hogonal matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "orthog(A (t)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "(b) The de rivative of the matrix squared is the following " }{TEXT 450 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 451 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 452 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "`der(A^2)` := map(combine, map(diff, evalm(A (t)^2), t)) : Diff(['A(t)']^2, t) = matrix(`der(A^2)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*$)7#-%\"AG6#%\"tG\"\"#\"\"\"F--% 'matrixG6#7$7$,$-%$sinG6#,$F-F.!\"#,$-%$cosGF8F:7$,$F " 0 "" {MPLTEXT 1 0 66 "`2A der(A)` := map (combine, evalm(2*A(t) &* map(diff, A(t), t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "2 * 'A(t)' * Diff('A(t)', t) = matrix(`2A der (A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&-%\"AG6#%\"tG\"\"\"-%% DiffG6$F&F)F*\"\"#-%'matrixG6#7$7$,$-%$sinG6#,$F)F.!\"#,$-%$cosGF7F97$ ,$F;F.F4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 53 "Both matrices of (b) and (c) are equal by inspecti on." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 283 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 376 4 "N.B." }{TEXT -1 17 " If the matri x [" }{TEXT 377 1 "A" }{TEXT 379 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 378 1 ")" }{TEXT -1 8 "] is a " }{TEXT 381 12 "non-singular" } {TEXT -1 131 " matrix, then the derivative of the matrix inverse is e qual to the negative product of the matrix inverse, derivative, and in verse" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff(Inv(A(t)),t) = \+ -Inv(A(t))*Diff(A(t),t)*Inv(A(t))" "6#/-%%DiffG6$-%$InvG6#-%\"AG6#%\"t GF-,$*(-F(6#-F+6#F-\"\"\"-F%6$-F+6#F-F-F4-F(6#-F+6#F-F4!\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 456 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 457 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 458 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 382 1 "A" }{TEXT 384 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 383 1 ")" }{TEXT -1 10 "] given as " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "A(t) := matrix(2, 2, [1 , t, 2, t^2]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"AG6#%\"tG-%'matrixG6#7$7$\"\"\"F'7$\"\"#*$)F'F/F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "( a) The derivative of the inverse of [" }{TEXT 385 1 "A" }{TEXT 387 1 " (" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 386 1 ")" }{TEXT -1 20 "] is th e following " }{TEXT 459 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 460 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 461 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "`der(in v(A))` := map(simplify, map(diff, inverse(A(t)), t)) : Diff(Inv('A(t )'), t) = matrix(`der(inv(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%%DiffG6$-%$InvG6#-%\"AG6#%\"tGF--%'matrixG6#7$7$,$*&\"\"\"F5*$),&F-F 5\"\"#!\"\"F9F5F:!\"#F47$,$*&,&F-F5F5F:F5*&)F-F9F5F7F5F:\"\"%,$F>F;" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "(b) The negative product of the inverse of [" }{TEXT 388 1 "A" }{TEXT 390 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 389 1 ")" } {TEXT -1 18 "], derivative of [" }{TEXT 391 1 "A" }{TEXT 393 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 392 1 ")" }{TEXT -1 19 "], and inve rse of [" }{TEXT 394 1 "A" }{TEXT 396 1 "(" }{XPPEDIT 18 0 "t" "6#%\"t G" }{TEXT 395 1 ")" }{TEXT -1 20 "] is the following " }{TEXT 462 1 " (" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 463 3 " \327 " }{XPPEDIT 18 0 " 2" "6#\"\"#" }{TEXT 464 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`-inv(A) der(A) inv(A)` := map(simplify, evalm(-inverse(A(t)) &* map(diff, A(t), t) &* inverse(A(t)))) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "`\226Inv`('A(t)') * Diff('A( t)', t) * Inv('A(t)') = matrix(`-inv(A) der(A) inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%%|atInvG6#-%\"AG6#%\"tG\"\"\"-%%DiffG6$F(F +F,-%$InvGF'F,-%'matrixG6#7$7$,$*&F,F,*$),&F+F,\"\"#!\"\"F" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 507 5 "equal" }{TEXT -1 64 " function applied to the resultant matrice s of (a) and (b), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "e qual(`der(inv(A))`, `-inv(A) der(A) inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "returns the " }{TEXT 571 7 "Boole an" }{TEXT -1 8 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 47 ", w hich verifies that both matrices are equal." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 380 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 4 "N.B." }{TEXT -1 23 " The derivative of a " }{TEXT 289 8 "cons tant" }{TEXT -1 10 " matrix [" }{TEXT 284 1 "K" }{TEXT -1 10 "] is th e " }{TEXT 532 4 "zero" }{TEXT -1 26 " matrix of the same order" }}} {EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff(K,t)" "6#-%%DiffG6$%\"KG% \"tG" }{TEXT 465 3 " = " }{TEXT -1 1 "[" }{TEXT 285 1 "0" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 466 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 467 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 468 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 286 1 "K" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "K := matr ix(2, 3, [1, b, a^2, a, b^2, c]) : K = matrix(K) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"KG-%'matrixG6#7$7%\"\"\"%\"bG*$)%\"aG\"\"#F*7%F.* $)F+F/F*%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The derivative of [" }{TEXT 287 1 "K" } {TEXT -1 20 "] is the following " }{TEXT 469 1 "(" }{XPPEDIT 18 0 "2 " "6#\"\"#" }{TEXT 470 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 471 1 ")" }{TEXT -1 2 " " }{TEXT 574 4 "zero" }{TEXT -1 9 " matrix: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`der(K)` := map(diff, K , t) : Diff(K, t) = matrix(`der(K)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%\"KG%\"tG-%'matrixG6#7$7%\"\"!F.F.F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 288 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 4 "N.B." }{TEXT -1 9 " If two " }{TEXT 291 6 "square" }{TEXT -1 1 " " }{TEXT 292 12 "non-singular" }{TEXT -1 12 " matrices, \+ [" }{TEXT 293 1 "A" }{TEXT 351 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 350 1 ")" }{TEXT -1 7 "] and [" }{TEXT 294 1 "B" }{TEXT 353 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 352 1 ")" }{TEXT -1 322 "], are \+ differentiable in some common interval, the derivative of the inverse \+ of their product is equal to the product of the derivative of the inve rse of the post-multiplier and the inverse of the pre-multiplier plus \+ the product of the inverse of the post-multiplier and the derivative o f the inverse of the pre-multiplier" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff(Inv(A(t)*B(t)),t) = Diff(Inv(B(t)),t) * Inv(A(t)) \+ + Inv(B(t)) * Diff(Inv(A(t)),t)" "6#/-%%DiffG6$-%$InvG6#*&-%\"AG6#%\"t G\"\"\"-%\"BG6#F.F/F.,&*&-F%6$-F(6#-F16#F.F.F/-F(6#-F,6#F.F/F/*&-F(6#- F16#F.F/-F%6$-F(6#-F,6#F.F.F/F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For example, consider " } {TEXT 472 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 473 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 474 1 ")" }{TEXT -1 12 " matrices \+ [" }{TEXT 295 1 "A" }{TEXT 355 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 354 1 ")" }{TEXT -1 7 "] and [" }{TEXT 296 1 "B" }{TEXT 357 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 356 1 ")" }{TEXT -1 10 "] given \+ as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "A(t) := matrix(2, 2, [t, 1, -1, t^2]) : B(t) := matrix(2, 2, [2, t^2, t^3, 1]) : 'A(t) ' = A(t) ; 'B(t)' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"A G6#%\"tG-%'matrixG6#7$7$F'\"\"\"7$!\"\"*$)F'\"\"#F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'matrixG6#7$7$\"\"#*$)F'F-\"\"\"7$*$) F'\"\"$F0F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "(a) The derivative of the inverse of the produc t of both matrices is the following " }{TEXT 475 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 476 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 477 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`der(inv(AB))` := map(sort, map(factor, map(diff, inv erse(evalm(A(t) &* B(t))), t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Diff(Inv('A(t)' * 'B(t)'), t) = matrix(`der(inv(AB))` ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$InvG6#*&-%\"AG6#% \"tG\"\"\"-%\"BGF-F/F.-%'matrixG6#7$7$\"\"!,$*&*$)F.\"\"%F/F/*$),&*$)F .\"\"&F/F/\"\"#!\"\"FCF/FD!\"&7$,$*&*$)F.FCF/F/*&),&F.F/F/F/FCF/),(*$F JF/F/F.FDF/F/FCF/FD!\"$*&,0*$)F.\"#5F/FB*&\"#9F/)F.\"\")F/F/*&FCF/)F. \"\"(F/F/*&FZF/FAF/F/*&FZF/)F.\"\"$F/FD*&\"\"'F/FJF/F/FF/FD" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 "(b) The product of the derivative of the inverse of the post-multiplier and the inverse of the pre-multiplier plus the pr oduct of the inverse of the post-multiplier and the derivative of the \+ inverse of the pre-multiplier is the following " }{TEXT 478 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 479 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 480 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "`der(inv(B)) inv(A) + inv(B) der(inv(A))` := map(sort, map(factor, evalm(map(diff, inverse(B(t)), t) &* inverse(A( t)) + inverse(B(t)) &* map(diff, inverse(A(t)), t)))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "Diff(Inv('B(t)'), t) * Inv('A(t)') + Inv('B(t)') * Diff(Inv('A(t)'), t) = matrix(`der(inv(B)) inv(A) + i nv(B) der(inv(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%%DiffG 6$-%$InvG6#-%\"BG6#%\"tGF/\"\"\"-F*6#-%\"AGF.F0F0*&F)F0-F'6$F1F/F0F0-% 'matrixG6#7$7$\"\"!,$*&*$)F/\"\"%F0F0*$),&*$)F/\"\"&F0F0\"\"#!\"\"FIF0 FJ!\"&7$,$*&*$)F/FIF0F0*&),&F/F0F0F0FIF0),(*$FPF0F0F/FJF0F0FIF0FJ!\"$* &,0*$)F/\"#5F0FH*&\"#9F0)F/\"\")F0F0*&FIF0)F/\"\"(F0F0*&FjnF0FGF0F0*&F jnF0)F/\"\"$F0FJ*&\"\"'F0FPF0F0FBF0F0*(FRF0FTF0FDF0FJ" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 505 5 "equal" }{TEXT -1 64 " function applied to the result ant matrices of (a) and (b), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "equal(`der(inv(AB))`, `der(inv(B)) inv(A) + inv(B) de r(inv(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "r eturns the " }{TEXT 572 7 "Boolean" }{TEXT -1 9 " value " } {XPPMATH 20 "6#%%trueG" }{TEXT -1 47 ", which verifies that both matr ices are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 302 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 4 "N.B." }{TEXT -1 9 " If two " }{TEXT 298 6 "square" }{TEXT -1 1 " " }{TEXT 299 12 "non-singul ar" }{TEXT -1 12 " matrices, [" }{TEXT 300 1 "A" }{TEXT 359 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 358 1 ")" }{TEXT -1 7 "] and [" } {TEXT 301 1 "B" }{TEXT 361 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 360 1 ")" }{TEXT -1 332 "], are differentiable in some common interval , the derivative of the transpose of their product is equal to the pro duct of the derivative of the transpose of the post-multiplier and the transpose of the pre-multiplier plus the product of the transpose of \+ the post-multiplier and the derivative of the transpose of the pre-mul tiplier" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Diff(Transp(A(t)*B (t)),t) = Diff(Transp(B(t)),t) * Transp(A(t)) + Transp(B(t)) * Diff(Tr ansp(A(t)),t)" "6#/-%%DiffG6$-%'TranspG6#*&-%\"AG6#%\"tG\"\"\"-%\"BG6# F.F/F.,&*&-F%6$-F(6#-F16#F.F.F/-F(6#-F,6#F.F/F/*&-F(6#-F16#F.F/-F%6$-F (6#-F,6#F.F.F/F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "For example, consider the same " }{TEXT 481 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 482 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 483 1 ")" }{TEXT -1 12 " matrices \+ [" }{TEXT 303 1 "A" }{TEXT 363 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 362 1 ")" }{TEXT -1 7 "] and [" }{TEXT 304 1 "B" }{TEXT 365 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 364 1 ")" }{TEXT -1 12 "] as bef ore." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 86 "(a) The derivative of the transpose of the product of b oth matrices is the following " }{TEXT 484 1 "(" }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 485 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 486 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "`der(transp(AB))` := map(diff, transpose(evalm(A(t) & * B(t))), t) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Diff(Tran sp('A(t)' * 'B(t)'), t) = matrix(`der(transp(AB))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'TranspG6#*&-%\"AG6#%\"tG\"\"\"-%\"BG F-F/F.-%'matrixG6#7$7$,&\"\"#F/*&\"\"$F/)F.F8F/F/,$*$)F.\"\"%F/\"\"&7$ ,$*$F;F/F:\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "(b) The product of the derivative of the transpose of the post-multiplier and the transpose of the pre-multipl ier plus the product of the transpose of the post-multiplier and the d erivative of the transpose of the pre-multiplier is the following " } {TEXT 487 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 488 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 489 1 ")" }{TEXT -1 9 " matrix:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "`der(transp(B)) transp(A) + transp(B) der(transp(A))` := evalm(map(diff, transpose(B(t)), t) &* transpose(A(t)) + transpose(B(t)) &* map(diff, transpose(A(t)), t)) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "Diff(Transp('B(t)'), t ) * Transp('A(t)') + Transp('B(t)') * Diff(Transp('A(t)'), t) = matrix (`der(transp(B)) transp(A) + transp(B) der(transp(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%%DiffG6$-%'TranspG6#-%\"BG6#%\"tGF/\"\" \"-F*6#-%\"AGF.F0F0*&F)F0-F'6$F1F/F0F0-%'matrixG6#7$7$,&\"\"#F0*&\"\"$ F0)F/F>F0F0,$*$)F/\"\"%F0\"\"&7$,$*$FAF0F@\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 506 5 "equal" }{TEXT -1 64 " function applied to the resultant matrice s of (a) and (b), i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "e qual(`der(transp(AB))`, `der(transp(B)) transp(A) + transp(B) der(tran sp(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "r eturns the " }{TEXT 573 7 "Boolean" }{TEXT -1 9 " value " } {XPPMATH 20 "6#%%trueG" }{TEXT -1 47 ", which verifies that both matr ices are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 504 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (28) fo r \"" }{TEXT 496 39 "Limits of matrices comprising functions" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 495 67 "----------------- --------------------------------------------------" }}}}{MARK "271 0 0 " 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }