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1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 295 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 } {PSTYLE "Normal" -1 296 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" -1 297 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 }} {SECT 0 {EXCHG {PARA 271 "" 0 "" {TEXT 570 39 "MATRICES AND MATRIX OPE RATIONS: Unit 23" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 272 "" 0 "" {TEXT 572 23 "Dr. Wlodzislaw Kostecki" }}{PARA 273 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 274 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 275 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 276 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT 571 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 278 "" 0 "" {TEXT 573 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT 574 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 397 4 "(23)" }{TEXT -1 1 " " }{TEXT 257 30 "Functions of co nstant matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1208 10 "OBJECTIVES" }{TEXT 1209 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1210 1 "\225" }{TEXT -1 92 " To provide essential information concern ing expansion of functions of one variable using " }{TEXT 1212 7 "Tay lor\222" }{TEXT -1 8 "s and " }{TEXT 1211 10 "Maclaurin\222" }{TEXT -1 10 "s series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1213 1 "\225" } {TEXT -1 35 " To introduce the concepts of an " }{TEXT 1214 8 "inter val" }{TEXT -1 7 " and " }{TEXT 1215 6 "circle" }{TEXT -1 59 " of c onvergence of a series representing a function of a " }{TEXT 1224 4 " real" }{TEXT -1 7 " and " }{TEXT 1225 7 "complex" }{TEXT -1 25 " va riable, respectively." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1216 1 "\225" } {TEXT -1 105 " To provide definitions of the matrix series for real-v alued and complex-valued matrices, based on the " }{TEXT 1217 9 "Macl aurin" }{TEXT -1 9 " series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1218 1 "\225" }{TEXT -1 40 " To state the condition necessary for " }{TEXT 1219 10 "Maclaurin\222" }{TEXT -1 57 "s series of a square matrix to \+ be absolutely convergent." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1221 1 " \225" }{TEXT -1 33 " To introduce the concept of a " }{TEXT 1220 12 "well-defined" }{TEXT -1 23 " function of a matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1223 1 "\225" }{TEXT -1 64 " To provide a list of som e common functions together with the " }{TEXT 1262 23 "interval of co nvergence" }{TEXT -1 26 " of their corresponding " }{TEXT 1222 10 "M aclaurin\222" }{TEXT -1 24 "s series for reference." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1226 1 "\225" }{TEXT -1 123 " To provide a reference \+ example of computation of a trigonometric function of a real and compl ex variable using infinite " }{TEXT 1227 7 "Taylor\222" }{TEXT -1 10 "s series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1228 1 "\225" }{TEXT -1 42 " To introduce the only built-in function " }{TEXT 1229 11 "expone ntial" }{TEXT -1 57 " for computation of the exponential function of m atrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1230 1 "\225" }{TEXT -1 135 " To provide six step-by-step solved examples for computation of func tions of matrices with real and complex elements using truncated " } {TEXT 1231 10 "Maclaurin\222" }{TEXT -1 10 "s series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1232 1 "\225" }{TEXT -1 133 " To show how to pr e-process a matrix containing transcendental numbers to significantly \+ reduce the computation time of its function." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 1237 1 "\225" }{TEXT -1 42 " To examine six common functions \+ of the " }{TEXT 1238 4 "zero" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1239 1 "\225" }{TEXT -1 77 " To show how to obt ain the exact result of the exponential function of the " }{TEXT 1240 4 "unit" }{TEXT -1 56 " matrix and examine two other functions o f this matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1241 1 "\225" }{TEXT -1 51 " To examine seven cases of exponentiation of the " }{TEXT 1242 14 "imaginary unit" }{TEXT -1 95 " matrix multiplied by some sca lars involving the various multiplicities of the imaginary unit." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1246 1 "\225" }{TEXT -1 34 " To show how to make use of the " }{TEXT 1245 15 "Cayley-Hamilton" }{TEXT -1 46 " theorem for computation of matrix functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1243 1 "\225" }{TEXT -1 137 " To provide three step-by-ste p solved examples for computation of functions of matrices with real e lements using a method based on the " }{TEXT 1244 15 "Cayley-Hamilton " }{TEXT -1 54 " theorem and verify the results using the truncated \+ " }{TEXT 1253 10 "Maclaurin\222" }{TEXT -1 10 "s series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1247 1 "\225" }{TEXT -1 82 " To state the neces sary conditions that a matrix must satisfy in order that its " } {TEXT 1248 11 "square root" }{TEXT -1 6 " be " }{TEXT 1249 6 "unique " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1250 1 "\225" } {TEXT -1 70 " To provide two step-by-step solved examples for computa tion of the " }{TEXT 1251 11 "square root" }{TEXT -1 62 " of real an d complex matrices using the method based on the " }{TEXT 1252 15 "Ca yley-Hamilton" }{TEXT -1 54 " theorem and verify the results using th e truncated " }{TEXT 1254 10 "Maclaurin\222" }{TEXT -1 10 "s series. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1256 1 "\225" }{TEXT -1 191 " To de termine the conditions that a matrix must satisfy in order that its un ique square root could be computed in an alternative manner using the \+ functions natural logarithm and exponential." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 1257 1 "\225" }{TEXT -1 157 " To provide a number of variants of the various square matrices of second order that are each a non-un ique square root of the unit matrix of the same order." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1258 1 "\225" }{TEXT -1 188 " To provide two st ep-by-step solved examples for computation of functions of diagonal ma trices with real elements using similarity of matrices and verify the \+ results using the truncated " }{TEXT 1260 10 "Maclaurin\222" }{TEXT -1 10 "s series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1259 1 "\225" } {TEXT -1 205 " To provide four step-by-step solved examples for compu tation of functions of non-diagonal real matrices with distinct eigenv alues using similarity of matrices and verify the results using the tr uncated " }{TEXT 1261 10 "Maclaurin\222" }{TEXT -1 10 "s series." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "restart : with(linalg, augment, coldim, definite, \+ diag, eigenvals, eigenvects, exponential, inverse, jordan, matadd, row dim) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Functions of matrices are defined " }{TEXT 258 4 "on ly" }{TEXT -1 6 " for " }{TEXT 259 6 "square" }{TEXT -1 11 " matrice s." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "This Unit deals with computation of functions for " } {TEXT 832 8 "constant" }{TEXT -1 31 " matrices whose elements are " }{TEXT 833 4 "real" }{TEXT -1 5 " or " }{TEXT 834 7 "complex" }{TEXT -1 10 " numbers." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Consider a function of a real variable, \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 42 ", and a funct ion of a complex variable, " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 42 ", and assume that either function has a " }{TEXT 995 6 "Taylor" }{TEXT -1 37 " series expansion about the points " } {XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT 996 1 "," }{TEXT -1 24 " respectively, given by" }}}{EXCHG {PARA 291 "" 0 "" {XPPEDIT 18 0 "f (x)=Sum(a[n]*(x-x[0])^n,n=0..infinity)" "6#/-%\"fG6#%\"xG-%$SumG6$*&&% \"aG6#%\"nG\"\"\"),&F'F0&F'6#\"\"!!\"\"F/F0/F/;F5%)infinityG" }{TEXT -1 32 " and " }{XPPEDIT 18 0 "f(z)=Sum(a[n] *(z-z[0])^n,n=0..infinity)" "6#/-%\"fG6#%\"zG-%$SumG6$*&&%\"aG6#%\"nG \"\"\"),&F'F0&F'6#\"\"!!\"\"F/F0/F/;F5%)infinityG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Suppose th at there is an " }{TEXT 1021 8 "interval" }{TEXT -1 2 " " }{XPPEDIT 18 0 " [x[0]-r, x[0]+r]" "6#7$,&&%\"xG6#\"\"!\"\"\"%\"rG!\"\",&&F&6#F( F)F*F)" }{TEXT -1 29 " with the central point at " }{XPPEDIT 18 0 "x [0]" "6#&%\"xG6#\"\"!" }{TEXT -1 21 " and end points at " }{XPPEDIT 18 0 "-r" "6#,$%\"rG!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "r" "6 #%\"rG" }{TEXT 1001 1 "," }{TEXT -1 40 " within which the series repr esenting " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 46 " is absolutely convergent for all values of " }{XPPEDIT 18 0 "x" "6#%\"x G" }{TEXT 998 1 "," }{TEXT -1 43 " i.e. those which satisfy the inequ ality " }{XPPEDIT 18 0 "abs(x-x[0]) " 0 "" {MPLTEXT 1 0 100 "`cos(x)` := Su m((-1)^n*x^(2*n)/(2*n)!, n=0..infinity) : cos(x) = `cos(x)` ; Cos( x) := `cos(x)` :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"xG-%$ SumG6$*&*&)!\"\"%\"nG\"\"\")F',$F/\"\"#F0F0-%*factorialG6#F2F./F/;\"\" !%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 5 "Maple" }{TEXT -1 145 " evaluates this series to an internally adopted accuracy and returns the result with the number of digits determined by the environment variable " }{TEXT 267 6 "Digi ts" }{TEXT -1 18 " (the default is " }{XPPEDIT 18 0 "10" "6#\"#5" } {TEXT -1 22 "). For example, set " }{XPPEDIT 18 0 "x=2.7" "6#/%\"xG$ \"#F!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z= 2.7+1.2*I" "6#/%\" zG,&$\"#F!\"\"\"\"\"*&$\"#7F(F)%\"IGF)F)" }{TEXT -1 15 " and compute \+ " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\"zG" }{TEXT -1 6 " to " } {XPPEDIT 18 0 "15" "6#\"#:" }{TEXT -1 15 " digits. Thus," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "Digits := 15 : x := 2.7 : `cos (x)` := evalf(`cos(x)`) : x := 'x' : cos(x) = `cos(x)` ; z := 2. 7 + 1.2*I : `cos(z)` := evalf(subs(x=z, Cos(x))) : z := 'z' : co s(z) = `cos(z)` ; Digits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$cosG6#%\"xG$!0hq,U@2/*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$c osG6#%\"zG^$$!011sDjpj\"!#9$!0\\OH8M6X'!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "No such built-in \+ procedures exist in " }{TEXT 263 5 "Maple" }{TEXT -1 21 " for computat ion of " }{TEXT 341 6 "matrix" }{TEXT -1 39 " functions. The only ex ception is the " }{TEXT 339 11 "exponential" }{TEXT -1 43 " function, \+ which is accessible through the " }{TEXT 340 6 "linalg" }{TEXT -1 9 " \+ package." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 211 "Therefore, where computation of a matrix function is involved, the series must be truncated. The number of series terms adopted should result from a compromise between the computation time \+ and required accuracy." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "It is appropriate to list at this point some common functions together with the interval of convergence of their corresponding " }{TEXT 404 10 "Maclaurin\222" }{TEXT -1 152 "s series, which will be a good indication whether the sought-for function can be computed for a given matrix once its eigenvalues have been determined." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Although functions of a real variable " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 80 " are shown together with t he convergence interval of their series, substitute " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "z" "6#%\"zG" } {TEXT -1 17 " and the term \"" }{TEXT 1015 23 "interval of convergen ce" }{TEXT -1 10 "\" with \"" }{TEXT 1016 21 "circle of convergence " }{TEXT -1 115 "\" where complex-numbered matrices are involved. If \+ a real-valued matrix has complex eigenvalues, the designation " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " still holds but the " }{TEXT 328 21 "circle of convergence" }{TEXT -1 18 " comes in to play." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 52 "(1) Functions whose series is convergent for every " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 270 1 "," }{TEXT -1 36 " i.e. having convergence interval " }{XPPEDIT 18 0 "abs(x)1" "6#2\"\"\"-% $absG6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 1 "\225" }{TEXT -1 2 " " }{XPPEDIT 18 0 "arccoth(x)" "6#-%(arccothG6#%\"xG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "( 6) Function with convergence interval " }{XPPEDIT 18 0 "x>1" "6#2\"\" \"%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 1 "\225" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "arccosh(x)" "6#-%(arccoshG6#%\"xG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "(7) Fun ction with convergence interval " }{XPPEDIT 18 0 "0" "6#\"\"!" } {TEXT 286 3 " < " }{XPPEDIT 18 0 "x <= 2" "6#1%\"xG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 1 "\225" }{TEXT -1 2 " " }{XPPEDIT 18 0 "ln (x)" "6#-%#lnG6#%\"xG" }{TEXT -1 26 " as given by the series " } {XPPEDIT 18 0 "ln(x) = Sum((-1)^(n+1)/n*(x-1)^n, n=1..infinity)" "6#/- %#lnG6#%\"xG-%$SumG6$*(),$\"\"\"!\"\",&%\"nGF.F.F.F.F1F/),&F'F.F.F/F1F ./F1;F.%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 413 56 "Note that there are also other series re presentations of" }{TEXT -1 2 " " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#% \"xG" }{TEXT -1 2 " " }{TEXT 414 54 "with their respective, different convergence intervals" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Apart from comput ation of matrix functions based on the " }{TEXT 327 9 "Maclaurin" } {TEXT -1 137 " series representation of a given function, there are t wo other methods that may be used for this purpose. One of them is bas ed on the " }{TEXT 329 15 "Cayley-Hamilton" }{TEXT -1 49 " theorem [ refer to Unit (21)] and the other on " }{TEXT 1197 22 "similarity of \+ matrices" }{TEXT -1 23 " [refer to Unit (22)]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "All the thre e methods together with illustrating examples are presented in Section s " }{TEXT 1017 1 "A" }{TEXT -1 2 ", " }{TEXT 1018 1 "B" }{TEXT -1 6 " , and " }{TEXT 1019 1 "C" }{TEXT -1 14 " of this Unit." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 3 " A. " }{TEXT 493 58 "Computation of matrix functions using the Maclauri n series" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 111 "Below presented are examples for computation of s everal different matrix functions with the use of this method." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 9 "Example 1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Compute the " }{TEXT 886 17 "natural log arithm" }{TEXT -1 6 " of " }{TEXT 535 1 "(" }{XPPEDIT 18 0 "2" "6#\" \"#" }{TEXT 260 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 536 1 " )" }{TEXT -1 12 " matrices [" }{TEXT 261 1 "A" }{TEXT -1 7 "] and [" }{TEXT 330 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 106 "A := matrix(2, 2, [2, 3, 0, 2]) : B := matrix(2, 2, [1, -2, 1, 2]) : A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"#\"\"$7$\"\"!F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\"\"\"!\"#7$F*\" \"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 294 6 "Step 1" }{TEXT -1 40 ". Find the eigenvalues of either \+ matrix:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 343 1 "\225" }{TEXT -1 14 " for matrix [" }{TEXT 344 1 "A " }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroo ts(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$\"\"#F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "S ince neither of the characteristic roots is greater than " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 17 ", the function " }{TEXT 306 3 "ln( " }{TEXT -1 1 "[" }{TEXT 305 1 "A" }{TEXT -1 1 "]" }{TEXT 537 1 ")" } {TEXT -1 54 " is well defined for this matrix and may be computed." } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 345 1 "\225" }{TEXT -1 14 " for matrix [" }{TEXT 346 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$,&#\"\"$\"\"#\"\"\"*&^##F- F,F--%%sqrtG6#\"\"(F-F-,&F*F-*&^##!\"\"F,F-F1F-F-" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "root 1(B) := charroots(B)[1] : char_root[1](B) = root1(B) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\"BG,&#\"\"$\"\"#F(*&^ ##F(F.F(-%%sqrtG6#\"\"(F(F(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`abs(root1(B))` := abs(roo t1(B)) : Abs(char_root[1](B)) = `abs(root1(B))` ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$AbsG6#-&%*char_rootG6#\"\"\"6#%\"BG\"\"#" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Since the modulus of neither characteristic root is greater tha n " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 17 ", the function " } {TEXT 539 3 "ln(" }{TEXT -1 1 "[" }{TEXT 538 1 "A" }{TEXT -1 1 "]" } {TEXT 540 1 ")" }{TEXT -1 54 " is well defined for this matrix and ma y be computed." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 6 "Step 2" }{TEXT -1 30 ". Write the express ion for a " }{TEXT 405 10 "Maclaurin\222" }{TEXT -1 37 "s series rep resenting the function " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" } {TEXT -1 36 ", substitute the matrix names for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 27 ", and replace subtrahend " }{XPPEDIT 18 0 "1 " "6#\"\"\"" }{TEXT -1 36 " with the name of the unit matrix [" } {TEXT 910 1 "U" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`ln(A)` := subs((x-1)=(A-U), Sum((-1)^(n+1)*(x-1)^n/n, n=1..in finity)) : ln(A) = `ln(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%# lnG6#%\"AG-%$SumG6$*&*&)!\"\",&%\"nG\"\"\"F1F1F1),&F'F1%\"UGF.F0F1F1F0 F./F0;F1%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`ln(B)` := subs((x-1)=(B-U), Sum((-1)^(n+1)*(x-1)^n/n, n=1..infinity)) : ln(B) = `ln(B)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"BG-%$SumG6$*&*&)!\"\",&%\" nG\"\"\"F1F1F1),&F'F1%\"UGF.F0F1F1F0F./F0;F1%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 6 "S tep 3" }{TEXT -1 40 ". Bearing in mind that the unit matrix [" }{TEXT 899 1 "U" }{TEXT -1 38 "] appropriately sized for this case is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U := diag(1, 1) : U = matr ix(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 "" {TEXT -1 71 "substitute either matrix with its numeric al values and the unit matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "`ln(A)` := subs(A=matrix(A), U=matrix(U), `ln(A)`) : ln(A) = `ln(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"AG-%$SumG6$* &*&)!\"\",&%\"nG\"\"\"F1F1F1),&-%'matrixG6#7$7$\"\"#\"\"$7$\"\"!F9F1-F 56#7$7$F1F<7$F " 0 "" {MPLTEXT 1 0 72 "`ln(B) ` := subs(B=matrix(B), U=matrix(U), `ln(B)`) : ln(B) = `ln(B)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"BG-%$SumG6$*&*&)!\"\",&%\" nG\"\"\"F1F1F1),&-%'matrixG6#7$7$F1!\"#7$F1\"\"#F1-F56#7$7$F1\"\"!7$F@ F1F.F0F1F1F0F./F0;F1%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 6 "Step 4" }{TEXT -1 83 ". Eval uate the function of either matrix after truncating the series to the \+ first " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 8 " terms:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "`ln(A)` := evalf(evalm(sum(( -1)^(n+1)*(A - U)^n/n, n=1..50))) : ln(A) = matrix(`ln(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"AG-%'matrixG6#7$7$$\"+1;ZK o!#5$\"\"!F17$F0F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "`ln(B)` := evalf(evalm(sum((-1)^(n+ 1)*(B - U)^n/n, n=1..50))) : ln(B) = matrix(`ln(B)`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"BG-%'matrixG6#7$7$$!+6X+tK!\"&$!+-zW 4o!\"%7$$\"+^Rs/MF2$\"++NUxIF2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 9 "Example 2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Compute t he " }{TEXT 887 6 "cosine" }{TEXT -1 6 " of " }{TEXT 541 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 299 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 542 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 300 1 "A " }{TEXT -1 7 "] and [" }{TEXT 331 1 "Z" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "A := matrix(2, 2, [2, 4, 1, \+ 2]) : Z := matrix(2, 2, [1+5*I, 2-4*I, 3+2*I, 1-2*I]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; Z = matrix(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"#\"\"%7$\" \"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"ZG-%'matrixG6#7$7$^$\"\" \"\"\"&^$\"\"#!\"%7$^$\"\"$F.^$F+!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 301 6 "Step 1" }{TEXT -1 40 ". Find the eigenvalues of either matrix:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 347 1 "\225" } {TEXT -1 14 " for matrix [" }{TEXT 348 1 "A" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%+char_rootsG6#%\"AG6$\"\"!\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 349 1 "\225" }{TEXT -1 14 " f or matrix [" }{TEXT 350 1 "Z" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(Z) := eigenvals(Z) : char_roots(Z) = charroots(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#% \"ZG6$,&^$\"\"\"#\"\"$\"\"#F+*&#F+F.F+-%%sqrtG6#^$\"\"(!#KF+F+,&F*F+*& #F+F.F+*$F1F+F+!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "root1(Z) := charroots(Z)[1] \+ : char_root[1](Z) = root1(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- &%*char_rootG6#\"\"\"6#%\"ZG,&^$F(#\"\"$\"\"#F(*&#F(F/F(-%%sqrtG6#^$\" \"(!#KF(F(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`abs(root1(Z))` := evalf(abs(root1(Z))) : \+ Abs(char_root[1](Z)) = `abs(root1(Z))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AbsG6#-&%*char_rootG6#\"\"\"6#%\"ZG$\"+e7kUK!\"*" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Note that this step is " }{TEXT 325 3 "not" }{TEXT -1 32 " necessary since the series of " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6 #%\"xG" }{TEXT -1 21 " is convergent for " }{TEXT 543 3 "all" } {TEXT -1 2 " " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 304 1 "," }{TEXT -1 22 " which implies that " }{TEXT 303 4 "cos(" }{TEXT -1 1 "[" } {TEXT 302 1 "A" }{TEXT -1 1 "]" }{TEXT 544 1 ")" }{TEXT -1 23 " is we ll defined for " }{TEXT 415 5 "every" }{TEXT -1 43 " square matrix. \+ Similarly, the series of " }{XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\"zG " }{TEXT -1 21 " is convergent for " }{TEXT 416 3 "all" }{TEXT -1 2 " " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 332 1 "," }{TEXT -1 22 " whi ch implies that " }{TEXT 334 4 "cos(" }{TEXT -1 1 "[" }{TEXT 333 1 "Z " }{TEXT -1 1 "]" }{TEXT 545 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 417 5 "every" }{TEXT -1 31 " complex-valued square matrix. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 307 6 "Step 2" }{TEXT -1 30 ". Write the expression for a " } {TEXT 406 10 "Maclaurin\222" }{TEXT -1 37 "s series representing the \+ function " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\"zG" }{TEXT -1 35 " and \+ substitute the matrix names [" }{TEXT 918 1 "A" }{TEXT -1 7 "] and [" }{TEXT 919 1 "Z" }{TEXT -1 7 "] for " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 16 ", res pectively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`cos(A)` := s ubs(x=A, Sum((-1)^n*x^(2*n)/(2*n)!, n=0..infinity)) : cos(A) = `cos( A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"AG-%$SumG6$*&*&) !\"\"%\"nG\"\"\")F',$F/\"\"#F0F0-%*factorialG6#F2F./F/;\"\"!%)infinity G" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 88 "`cos(Z)` := subs(x=Z, Sum((-1)^n*x^(2*n)/(2*n)!, n= 0..infinity)) : cos(Z) = `cos(Z)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"ZG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")F',$F/\"\"#F0F0-%*fac torialG6#F2F./F/;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 902 4 "N.B." }{TEXT -1 2 " " } {TEXT 904 5 "Maple" }{TEXT -1 62 " returns the first term of either se ries as the scalar value " }{XPPMATH 20 "6#%$oneG" }{TEXT -1 94 ", a ccording to the result of the following computation performed exemplar ily for the matrix [" }{TEXT 903 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "`A^0` := evalm(A^0) : A ^ `0` = ` A^0` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"AG%\"0G\"\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Therefore, still with [" }{TEXT 905 1 "A" }{TEXT -1 47 "] as an example, the corresponding series for " }{XPPEDIT 18 0 "n=0" "6#/%\" nG\"\"!" }{TEXT -1 13 " to, e.g., " }{XPPEDIT 18 0 "n=5" "6#/%\"nG\" \"&" }{TEXT -1 22 " would be of the form" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "cos(A) = subs(x=A, sum((-1)^n*x^(2*n)/(2*n)!, n=0.. 5)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"AG,.\"\"\"F)*&#F )\"\"#F)*$)F'F,F)F)!\"\"*&#F)\"#CF))F'\"\"%F)F)*&#F)\"$?(F)*$)F'\"\"'F )F)F/*&#F)\"&?.%F))F'\"\")F)F)*&#F)\"(+)GOF)*$)F'\"#5F)F)F/" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "in which the first term is the algebraic number " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "However, adhering to th e common literature convention that [" }{TEXT 911 1 "A" }{TEXT -1 1 " ]" }{TEXT 914 1 "^" }{TEXT -1 2 "0 " }{TEXT 913 1 "=" }{TEXT -1 2 " [ " }{TEXT 912 1 "U" }{TEXT -1 60 "], the first term of the series shou ld be the unit matrix [" }{TEXT 915 1 "U" }{TEXT -1 79 "]. To obtain t his result, the original expression for the series is modified by" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "(a) replacing " }{XPPEDIT 18 0 "n =0" "6#/%\"nG\"\"!" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "n=1" "6#/% \"nG\"\"\"" }{TEXT -1 9 " in the " }{TEXT 1074 3 "sum" }{TEXT -1 14 " function, and" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(b) adding the \+ unit matrix [" }{TEXT 916 1 "U" }{TEXT -1 72 "] as the first term of t he series representation of the matrix function." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Consequent ly, the following expressions are adopted in this case:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`cos(A)` := U + subs(x=A, Sum((-1)^ n*x^(2*n)/(2*n)!, n=1..infinity)) : cos(A) = `cos(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"AG,&%\"UG\"\"\"-%$SumG6$*&*&)!\"\" %\"nGF*)F',$F2\"\"#F*F*-%*factorialG6#F4F1/F2;F*%)infinityGF*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`cos(Z)` := U + subs(x=Z, Sum((-1)^n*x^(2*n)/(2*n)!, \+ n=1..infinity)) : cos(Z) = `cos(Z)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"ZG,&%\"UG\"\"\"-%$SumG6$*&*&)!\"\"%\"nGF*)F',$F2\" \"#F*F*-%*factorialG6#F4F1/F2;F*%)infinityGF*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 6 "Step 3" } {TEXT -1 114 ". With the unit matrix as in Example 1, substitute the m atrix names with their corresponding numerical structures:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`cos(A)` := subs(U=matrix(U), A=mat rix(A), `cos(A)`) : cos(A) = `cos(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"AG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F/F.F.-%$ SumG6$*&*&)!\"\"%\"nGF.)-F*6#7$7$\"\"#\"\"%7$F.F>,$F8F>F.F.-%*factoria lG6#FAF7/F8;F.%)infinityGF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`cos(Z)` := subs(U=matrix( U), Z=matrix(Z), `cos(Z)`) : cos(Z) = `cos(Z)` ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$cosG6#%\"ZG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F/F.F.- %$SumG6$*&*&)!\"\"%\"nGF.)-F*6#7$7$^$F.\"\"&^$\"\"#!\"%7$^$\"\"$FA^$F. !\"#,$F8FAF.F.-%*factorialG6#FHF7/F8;F.%)infinityGF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 309 6 "Step 4 " }{TEXT -1 83 ". Evaluate the function of either matrix after truncat ing the series to the first " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 8 " terms:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "`cos(A)` := \+ evalf(evalm(U + sum((-1)^n*A^(2*n)/(2*n)!, n=1..50))) : cos(A) = mat rix(`cos(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"AG-%'m atrixG6#7$7$$\"+'*=yJ " 0 "" {MPLTEXT 1 0 97 "` cos(Z)` := evalf(evalm(U + sum((-1)^n*Z^(2*n)/(2*n)!, n=1..50))) : c os(Z) = matrix(`cos(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG 6#%\"ZG-%'matrixG6#7$7$^$$\"+c)fv#**!\"*$\"+j?0`&)F0^$$!+`EL&3)F0$!+&> ^.A(F07$^$$\"+@U\"[e'F0$!+.5BYdF0^$$!+&p^hW'F0$\"+YKH/TF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 310 9 "E xample 3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 13 "Compute the " }{TEXT 888 11 "exponential" }{TEXT -1 19 " function of the " }{TEXT 546 1 "(" }{XPPEDIT 18 0 "2" "6#\" \"#" }{TEXT 311 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 547 1 " )" }{TEXT -1 12 " matrices [" }{TEXT 312 1 "A" }{TEXT -1 7 "] and [" }{TEXT 335 1 "Z" }{TEXT -1 18 "] given as before." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 6 "Step 1" } {TEXT -1 40 ". Find the eigenvalues of either matrix:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "T his step is " }{TEXT 326 3 "not" }{TEXT -1 32 " necessary since the se ries of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 21 " \+ is convergent for " }{TEXT 548 3 "all" }{TEXT -1 2 " " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 315 1 "," }{TEXT -1 22 " which implies that \+ " }{TEXT 316 4 "exp(" }{TEXT -1 1 "[" }{TEXT 314 1 "A" }{TEXT -1 1 "] " }{TEXT 549 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 418 5 "every" }{TEXT -1 42 " square matrix. Likewise, the series of " } {XPPEDIT 18 0 "exp(z)" "6#-%$expG6#%\"zG" }{TEXT -1 21 " is convergen t for " }{TEXT 419 3 "all" }{TEXT -1 2 " " }{XPPEDIT 18 0 "z" "6#%\" zG" }{TEXT 337 1 "," }{TEXT -1 22 " which implies that " }{TEXT 336 4 "exp(" }{TEXT -1 1 "[" }{TEXT 338 1 "Z" }{TEXT -1 1 "]" }{TEXT 550 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 420 5 "every" } {TEXT -1 47 " square matrix with complex-numbered elements." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 6 "Step 2" }{TEXT -1 30 ". Write the expression for a " }{TEXT 889 10 "Maclaurin\222" }{TEXT -1 37 "s series representing the functi on " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "exp(z)" "6#-%$expG6#%\"zG" }{TEXT -1 35 " and subst itute the matrix names [" }{TEXT 920 1 "A" }{TEXT -1 7 "] and [" } {TEXT 921 1 "Z" }{TEXT -1 7 "] for " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 16 ", res pectively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`exp(A)` := s ubs(x=A, Sum(x^n/n!, n=0..infinity)) : exp(A) = `exp(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"AG-%$SumG6$*&)F'%\"nG\"\"\"-%*f actorialG6#F-!\"\"/F-;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`exp(Z)` := subs(x=Z, Sum(x^n/n!, n=0..infinity)) : exp(Z) = `exp(Z)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"ZG-%$SumG6$*&)F'%\"nG\"\" \"-%*factorialG6#F-!\"\"/F-;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 900 4 "N.B." }{TEXT -1 21 " Like in Example 2, " }{TEXT 901 5 "Maple" }{TEXT -1 62 " retu rns the first term of either series as the scalar value " }{XPPMATH 20 "6#%$oneG" }{TEXT -1 20 ". With the matrix [" }{TEXT 906 1 "A" } {TEXT -1 47 "] as an example, the corresponding series for " } {XPPEDIT 18 0 "n=0" "6#/%\"nG\"\"!" }{TEXT -1 13 " to, e.g., " } {XPPEDIT 18 0 "n=5" "6#/%\"nG\"\"&" }{TEXT -1 22 " would be of the fo rm" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "exp(A) = subs(x=A, su m(x^n/n!, n=0..5)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"A G,.\"\"\"F)F'F)*&#F)\"\"#F))F'F,F)F)*&#F)\"\"'F))F'\"\"$F)F)*&#F)\"#CF ))F'\"\"%F)F)*&#F)\"$?\"F))F'\"\"&F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "For the reasons s et forth in Example 2, the following expressions are adopted in this c ase:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`exp(A)` := U + sub s(x=A, Sum(x^n/n!, n=1..infinity)) : exp(A) = `exp(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"AG,&%\"UG\"\"\"-%$SumG6$*&)F'% \"nGF*-%*factorialG6#F0!\"\"/F0;F*%)infinityGF*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`exp(Z )` := U + subs(x=Z, Sum(x^n/n!, n=1..infinity)) : exp(Z) = `exp(Z)` \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"ZG,&%\"UG\"\"\"-%$Su mG6$*&)F'%\"nGF*-%*factorialG6#F0!\"\"/F0;F*%)infinityGF*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 318 6 "S tep 3" }{TEXT -1 114 ". With the unit matrix as in Example 1, substitu te the matrix names with their corresponding numerical structures:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`exp(A)` := subs(U=matrix(U ), A=matrix(A), `exp(A)`) : exp(A) = `exp(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"AG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F/F.F.-%$ SumG6$*&)-F*6#7$7$\"\"#\"\"%7$F.F:%\"nGF.-%*factorialG6#F=!\"\"/F=;F.% )infinityGF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`exp(Z)` := subs(U=matrix(U), Z=matrix(Z) , `exp(Z)`) : exp(Z) = `exp(Z)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$expG6#%\"ZG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F/F.F.-%$SumG6$*&)-F*6# 7$7$^$F.\"\"&^$\"\"#!\"%7$^$\"\"$F=^$F.!\"#%\"nGF.-%*factorialG6#FD!\" \"/FD;F.%)infinityGF." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 319 6 "Step 4" }{TEXT -1 83 ". Evaluate t he function of either matrix after truncating the series to the first \+ " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 8 " terms:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "`exp(A)` := evalf(evalm(U + sum(A^n /n!, n=1..50))) : exp(A) = matrix(`exp(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"AG-%'matrixG6#7$7$$\"+-v!*zF!\")$\"+.]\")f `F/7$$\"+^P&*R8F/F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "`exp(Z)` := evalf(evalm(U + \+ sum(Z^n/n!, n=1..50))) : exp(Z) = matrix(`exp(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"ZG-%'matrixG6#7$7$^$$\"+R:3Kg!\"*$\"+ 0;2w5!\")^$$\"+3a/,:F3$!+3JK88F37$^$$\"+f%Rb+*F0$\"+P'o@L\"F3^$$\"+\\b M&y\"F3$!+LDJ8=F3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 320 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 321 4 "N.B." }{TEXT -1 65 " T he exponential function of a square matrix may be computed in " } {TEXT 322 5 "Maple" }{TEXT -1 1 " " }{TEXT 323 8 "directly" }{TEXT -1 21 ", using the built-in " }{TEXT 324 11 "exponential" }{TEXT -1 51 " \+ function. For the matrices of Example 3, it yields" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`ex p(A)` := evalf(exponential(A)) : exp(A) = matrix(`exp(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"AG-%'matrixG6#7$7$$\"+-v! *zF!\")$\"+.]\")f`F/7$$\"+^P&*R8F/F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`exp(Z)` := evalf (exponential(Z)) : exp(Z) = matrix(`exp(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"ZG-%'matrixG6#7$7$^$$\"+-fH_a!\"*$\"+w4D?6 !\")^$$\"+4a/,:F3$!+2JK88F37$^$$\"+^%Rb+*F0$\"+Q'o@L\"F3^$$\"+()pOF " 0 "" {MPLTEXT 1 0 88 "A := matrix(2, 2, [1/3, -2/7, -1/6, 2/3]) : B \+ := matrix(2, 2, [1/3, -2/7, 1/6, 2/3]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$#\"\"\"\"\"$#!\"#\"\"(7$#!\"\" \"\"'#\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7 $#\"\"\"\"\"$#!\"#\"\"(7$#F+\"\"'#\"\"#F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 6 "Step 1" }{TEXT -1 40 ". Find the eigenvalues of either matrix:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 356 1 "\225" } {TEXT -1 14 " for matrix [" }{TEXT 357 1 "A" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%+char_rootsG6#%\"AG6$,&#\"\"\"\"\"#F+*&#F+\"#UF+-%%sqrtG6#\"$L\"F+ F+,&F*F+*&#F+F/F+*$F0F+F+!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "root1(A) := charroots(A )[1] : root2(A) := charroots(A)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1](A) = root1(A) ; char_root[2](A) = root 2(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\" AG,&#F(\"\"#F(*&#F(\"#UF(-%%sqrtG6#\"$L\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"#6#%\"AG,&#\"\"\"F(F-*&#F-\"#UF-*$ -%%sqrtG6#\"$L\"F-F-!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`abs(root1(A))` := evalf(abs (root1(A))) : `abs(root2(A))` := evalf(abs(root2(A))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Abs(char_root[1](A)) = `abs(root1(A ))` ; Abs(char_root[2](A)) = `abs(root2(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AbsG6#-&%*char_rootG6#\"\"\"6#%\"AG$\"+Q#[eu(!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AbsG6#-&%*char_rootG6#\"\"#6#%\"A G$\"+i<:aA!#5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Since either of the characteristic roots \+ is smaller than " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 17 ", the \+ function " }{TEXT 359 7 "arctan(" }{TEXT -1 1 "[" }{TEXT 358 1 "A" } {TEXT -1 1 "]" }{TEXT 553 1 ")" }{TEXT -1 54 " is well defined for th is matrix and may be computed." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 360 1 "\225" }{TEXT -1 14 " for mat rix [" }{TEXT 361 1 "B" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(B) := eigenvals(B) : char_roots(B) = char roots(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"BG6$ ,&#\"\"\"\"\"#F+*&^##F+\"#UF+-%%sqrtG6#\"#NF+F+,&F*F+*&^##!\"\"F0F+F1F +F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "root1(B) := charroots(B)[1] : root2(B) := charro ots(B)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1] (B) = root1(B) ; char_root[2](B) = root2(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\"BG,&#F(\"\"#F(*&^##F(\"#UF( -%%sqrtG6#\"#NF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6 #\"\"#6#%\"BG,&#\"\"\"F(F-*&^##!\"\"\"#UF--%%sqrtG6#\"#NF-F-" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`abs(root1(B))` := evalf(abs(root1(B))) : Abs(char_ root[1](B)) = `abs(root1(B))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $AbsG6#-&%*char_rootG6#\"\"\"6#%\"BG$\"+>[i%>&!#5" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Since the \+ modulus of either characteristic root is less than " }{XPPEDIT 18 0 " 1" "6#\"\"\"" }{TEXT -1 17 ", the function " }{TEXT 362 7 "arctan(" }{TEXT -1 1 "[" }{TEXT 363 1 "B" }{TEXT -1 1 "]" }{TEXT 554 1 ")" } {TEXT -1 54 " is well defined for this matrix and may be computed." } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 364 6 "Step 2" }{TEXT -1 30 ". Write the expression for a " } {TEXT 407 10 "Maclaurin\222" }{TEXT -1 37 "s series representing the \+ function " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 35 " and substitute the matrix names [" }{TEXT 922 1 "A" }{TEXT -1 7 "] and [" }{TEXT 923 1 "B" }{TEXT -1 7 "] for " }{XPPEDIT 18 0 " x" "6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "`arctan(A)` := subs(x=A, Sum((-1)^n*x^(2*n+1)/(2*n+1), n=0..infin ity)) : arctan(A) = `arctan(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%'arctanG6#%\"AG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")F',&F/\"\"#F0F0F0F0F 2F./F/;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "`arctan(B)` := subs(x=B, Su m((-1)^n*x^(2*n+1)/(2*n+1), n=0..infinity)) : arctan(B) = `arctan(B) ` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"BG-%$SumG6$*&*& )!\"\"%\"nG\"\"\")F',&F/\"\"#F0F0F0F0F2F./F/;\"\"!%)infinityG" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 365 6 "Step 3" }{TEXT -1 76 ". Substitute the matrix names with their \+ corresponding numerical structures:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`arctan(A)` := subs(A=matrix(A), `arctan(A)`) : arc tan(A) = `arctan(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6 #%\"AG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")-%'matrixG6#7$7$#F0\"\"$#!\"#\"\" (7$#F.\"\"'#\"\"#F8,&F/F@F0F0F0F0FAF./F/;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`arctan(B)` := subs(B=matrix(B), `arctan(B)`) : arctan(B) = `a rctan(B)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"BG-%$Su mG6$*&*&)!\"\"%\"nG\"\"\")-%'matrixG6#7$7$#F0\"\"$#!\"#\"\"(7$#F0\"\"' #\"\"#F8,&F/F@F0F0F0F0FAF./F/;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 366 6 "Step 4" } {TEXT -1 83 ". Evaluate the function of either matrix after truncating the series to the first " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 8 " terms:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`arctan(A)` := \+ evalf(evalm(sum((-1)^n*A^(2*n+1)/(2*n+1), n=0..50))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "arctan(A) = matrix(`arctan(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"AG-%'matrixG6#7$7$$ \"+$H@l2$!#5$!+InLvAF/7$$!+f(zsK\"F/$\"+533JdF/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`arcta n(B)` := evalf(evalm(sum((-1)^n*B^(2*n+1)/(2*n+1), n=0..50))) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "arctan(B) = matrix(`arctan(B )`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"BG-%'matrixG6 #7$7$$\"+*y#)eO$!#5$!+MS/)G#F/7$$\"+`BpM8F/$\"+'\\n_.'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 368 9 "E xample 5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 13 "Compute the " }{TEXT 891 11 "arc tangent" }{TEXT -1 6 " of " }{TEXT 555 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 369 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 556 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 370 1 "A" }{TEXT -1 7 "] and [" }{TEXT 371 1 "Z" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "A := matrix(2, 2, [0.2, -0.4, 0.1, 0.3]) : Z := ma trix(2, 2, [0.1+0.5*I, 0.2-0.4*I, 0.3+0.2*I, 0.1-0.2*I]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; Z = matrix(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$$\"\"#!\"\"$! \"%F,7$$\"\"\"F,$\"\"$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"ZG-%'m atrixG6#7$7$^$$\"\"\"!\"\"$\"\"&F-^$$\"\"#F-$!\"%F-7$^$$\"\"$F-F1^$F+$ !\"#F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 372 6 "Step 1" }{TEXT -1 40 ". Find the eigenvalues of eith er matrix:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 373 1 "\225" }{TEXT -1 14 " for matrix [" }{TEXT 374 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charr oots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"AG6$^$$\"+++++D!#5$\"+t; \\O>F,^$F*$!+t;\\O>F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "root1(A) := charroots(A)[1] \+ : root2(A) := charroots(A)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1](A) = root1(A) ; char_root[2](A) = root 2(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\" AG^$$\"+++++D!#5$\"+t;\\O>F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*c har_rootG6#\"\"#6#%\"AG^$$\"+++++D!#5$!+t;\\O>F." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`abs (root1(A))` := abs(root1(A)) : Abs(char_root[1](A)) = `abs(root1(A)) ` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$AbsG6#-&%*char_rootG6#\"\" \"6#%\"AG$\"+gwFiJ!#5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Since the modulus of either charac teristic root is less than " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 17 ", the function " }{TEXT 375 7 "arctan(" }{TEXT -1 1 "[" }{TEXT 376 1 "A" }{TEXT -1 1 "]" }{TEXT 557 1 ")" }{TEXT -1 54 " is well def ined for this matrix and may be computed." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 377 1 "\225" }{TEXT -1 14 " for matrix [" }{TEXT 378 1 "Z" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(Z) := eigenvals(Z) : char_roo ts(Z) = charroots(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_roo tsG6#%\"ZG6$^$$!+nlDH7!#5$\"+i,K%H$F,^$$\"+olDHKF,$!+-;?VH!#6" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "root1(Z) := charroots(Z)[1] : root2(Z) := charroots (Z)[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "char_root[1](Z) = root1(Z) ; char_root[2](Z) = root2(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"\"6#%\"ZG^$$!+nlDH7!#5$\"+i,K%H$F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%*char_rootG6#\"\"#6#%\"ZG^$$\" +olDHK!#5$!+-;?VH!#6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`abs(root1(Z))` := abs(root1 (Z)) : `abs(root2(Z))` := abs(root2(Z)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Abs(char_root[1](Z)) = `abs(root1(Z))` ; Abs(cha r_root[2](Z)) = `abs(root2(Z))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$AbsG6#-&%*char_rootG6#\"\"\"6#%\"ZG$\"+WN>;N!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$AbsG6#-&%*char_rootG6#\"\"#6#%\"ZG$\"+g7kUK!#5" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Since the modulus of either characteristic root is less t han " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 17 ", the function " }{TEXT 379 7 "arctan(" }{TEXT -1 1 "[" }{TEXT 380 1 "Z" }{TEXT -1 1 "] " }{TEXT 558 1 ")" }{TEXT -1 54 " is well defined for this matrix and may be computed." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 381 6 "Step 2" }{TEXT -1 30 ". Write the express ion for a " }{TEXT 408 10 "Maclaurin\222" }{TEXT -1 37 "s series rep resenting the function " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#% \"xG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "arctan(z)" "6#-%'arctanG6# %\"zG" }{TEXT -1 35 " and substitute the matrix names [" }{TEXT 924 1 "A" }{TEXT -1 7 "] and [" }{TEXT 925 1 "Z" }{TEXT -1 7 "] for " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z" " 6#%\"zG" }{TEXT -1 16 ", respectively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "`arctan(A)` := subs(x=A, Sum((-1)^n*x^(2*n+1)/(2*n+1 ), n=0..infinity)) : arctan(A) = `arctan(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"AG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")F',&F/ \"\"#F0F0F0F0F2F./F/;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "`arctan(Z) ` := subs(x=Z, Sum((-1)^n*x^(2*n+1)/(2*n+1), n=0..infinity)) : arcta n(Z) = `arctan(Z)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#% \"ZG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")F',&F/\"\"#F0F0F0F0F2F./F/;\"\"!%)i nfinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 382 6 "Step 3" }{TEXT -1 76 ". Substitute the matrix name s with their corresponding numerical structures:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`arctan(A)` := subs(A=matrix(A), `arctan(A)`) : arctan(A) = `arctan(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%' arctanG6#%\"AG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")-%'matrixG6#7$7$$\"\"#F.$ !\"%F.7$$F0F.$\"\"$F.,&F/F8F0F0F0F0F?F./F/;\"\"!%)infinityG" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`arctan(Z)` := subs(Z=matrix(Z), `arctan(Z)`) : arc tan(Z) = `arctan(Z)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6 #%\"ZG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")-%'matrixG6#7$7$^$$F0F.$\"\"&F.^$ $\"\"#F.$!\"%F.7$^$$\"\"$F.F<^$F8$!\"#F.,&F/F=F0F0F0F0FGF./F/;\"\"!%)i nfinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 383 6 "Step 4" }{TEXT -1 83 ". Evaluate the function of e ither matrix after truncating the series to the first " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 8 " terms:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "`arctan(A)` := evalm(sum((-1)^n*A^(2*n+1)/(2*n+1), n= 0..50)) : arctan(A) = matrix(`arctan(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"AG-%'matrixG6#7$7$$\"+P,jg?!#5$!+!3**)) z$F/7$$\"+,xC(\\*!#6$\"+2\\N5IF/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "`arctan(Z)` := evalm (sum((-1)^n*Z^(2*n+1)/(2*n+1), n=0..50)) : arctan(Z) = matrix(`arcta n(Z)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"ZG-%'matri xG6#7$7$^$$\"+')[HT))!#6$\"+.,Oy]!#5^$$\"+x&RA,#F3$!+CcoQSF37$^$$\"+UX sHIF3$\"+A-m8?F3^$$\"+(f\\=u)F0$!+suO%)>F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1023 9 "Example 6" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Compute the " }{TEXT 1029 11 "third power" }{TEXT -1 6 " of \+ " }{TEXT 1028 4 "sine" }{TEXT -1 8 " of a " }{TEXT 1026 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1024 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1027 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 1025 1 "A " }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := matrix(2, 2, [-3, 5, 1, 9]) : A = matrix(A) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$!\"$\"\"&7$\"\"\"\"\"*" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1030 6 "Step 1" }{TEXT -1 37 ". Find the eigenvalues of the matrix:" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "This step is " }{TEXT 1034 3 "not" }{TEXT -1 32 " necessa ry since the series of " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" } {TEXT -1 21 " is convergent for " }{TEXT 1036 3 "all" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 1032 1 "," }{TEXT -1 22 " whic h implies that " }{TEXT 1033 4 "sin(" }{TEXT -1 1 "[" }{TEXT 1031 1 " A" }{TEXT -1 1 "]" }{TEXT 1037 1 ")" }{TEXT -1 23 " is well defined f or " }{TEXT 1035 5 "every" }{TEXT -1 24 " square matrix. So is " } {TEXT 1053 5 "\{sin(" }{TEXT -1 1 "[" }{TEXT 1052 1 "A" }{TEXT -1 1 "] " }{TEXT 1054 4 ")\}^3" }{TEXT -1 25 " and it may be computed." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1038 6 "Step 2" }{TEXT -1 30 ". Write the expression for a " }{TEXT 1039 10 "Maclaurin\222" }{TEXT -1 37 "s series representing the funct ion " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 34 " and \+ substitute the matrix name [" }{TEXT 1040 1 "A" }{TEXT -1 7 "] for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`sin(A)` := subs(x=A, Sum((-1)^n*x^(2*n+1)/(2*n+1) !, n=0..infinity)) : sin(A) = `sin(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%\"AG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")F',&F/\" \"#F0F0F0F0-%*factorialG6#F2F./F/;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1041 6 "Step 3" }{TEXT -1 72 ". Substitute the matrix name with its corresponding nume rical structure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`sin(A) ` := subs(A=matrix(A), `sin(A)`) : sin(A) = `sin(A)` ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$sinG6#%\"AG-%$SumG6$*&*&)!\"\"%\"nG\"\"\")-% 'matrixG6#7$7$!\"$\"\"&7$F0\"\"*,&F/\"\"#F0F0F0F0-%*factorialG6#F;F./F /;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1042 6 "Step 4" }{TEXT -1 73 ". Evaluate the mat rix function after truncating the series to the first " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 8 " terms:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "`sin(A)` := evalf(evalm(sum((-1)^n*A^(2*n+1)/(2*n+1)! , n=0..50))) : sin(A) = matrix(`sin(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%\"AG-%'matrixG6#7$7$$\"+wy-6D!#5$!+=hr\\#*!# 67$$!+CK%*\\=F2$\"+r+'4\"HF2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1043 7 "Step 5." }{TEXT -1 48 " Comput e the third power of the matrix function:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "`(sin(A))^3` := evalm(`sin(A)` ^ 3) : (sin(A))^3 \+ = matrix(`(sin(A))^3`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%$sin G6#%\"AG\"\"$\"\"\"-%'matrixG6#7$7$$\"+TV=u;!#6$!+k3&\\u'!#77$$!+t,**[ 8F6$\"+KLhRb!#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1045 4 "N.B." }{TEXT -1 9 " Since " }{TEXT 1044 10 "Maclaurin\222" }{TEXT -1 42 "s series representation of the \+ function " }{XPPEDIT 18 0 "sin(x)^3" "6#*$-%$sinG6#%\"xG\"\"$" } {TEXT -1 16 " is known, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "sin(x)^3 = (3/4)*Sum((-1)^n*(1-9^n)*x^(2*n+1)/(2*n+1)!, n=0..inf inity) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%$sinG6#%\"xG\"\"$\" \"\",$-%$SumG6$*&*()!\"\"%\"nGF+,&F+F+)\"\"*F4F3F+)F),&F4\"\"#F+F+F+F+ -%*factorialG6#F9F3/F4;\"\"!%)infinityG#F*\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "it may be us ed for direct computation of the function " }{TEXT 1050 5 "\{sin(" } {TEXT -1 1 "[" }{TEXT 1049 1 "A" }{TEXT -1 1 "]" }{TEXT 1051 4 ")\}^3 " }{TEXT -1 38 " by following similar steps as above." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 " Evaluation of the matrix function after \+ truncating the series to the first " }{XPPEDIT 18 0 "50" "6#\"#]" } {TEXT -1 25 " terms yields the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`(sin(A))^3` := evalf(evalm((3/4)*sum((-1)^n*(1-9^n)* A^(2*n+1)/(2*n+1)!, n=0..50))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "(sin(A))^3 = matrix(`(sin(A))^3`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%$sinG6#%\"AG\"\"$\"\"\"-%'matrixG6#7$7$$\"+SV= u;!#6$!+i3&\\u'!#77$$!+s,**[8F6$\"+JLhRb!#8" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "which is practi cally equal to the matrix obtained by computing the third power of " }{TEXT 1047 4 "sin(" }{TEXT -1 1 "[" }{TEXT 1046 1 "A" }{TEXT -1 1 "] " }{TEXT 1048 1 ")" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT 1055 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1057 4 " N.B." }{TEXT -1 91 " Various combinations of matrix functions may als o be computed using the method based on " }{TEXT 1056 10 "Maclaurin \222" }{TEXT -1 210 "s series representation of the functions concern ed. Where each function in a combination has different convergence int erval (or circle), the resultant function will be convergent within th e narrowest interval." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Exemplarily, consider the product \+ " }{XPPEDIT 18 0 "exp(x)*cos(x)" "6#*&-%$expG6#%\"xG\"\"\"-%$cosG6#F' F(" }{TEXT -1 64 " where the series for either function is convergent for every " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 1059 1 "," }{TEXT -1 33 " i.e. has convergence interval " }{XPPEDIT 18 0 "abs(x)