{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Geneva" 1 18 0 0 0 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 258 "Geneva" 1 12 0 0 0 1 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 266 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 273 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 274 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 282 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "Geneva" 1 12 0 0 0 1 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 289 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 290 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 297 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Geneva" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ge neva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Geneva" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Mo naco" 1 9 0 0 255 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Geneva" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 1 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 259 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 30 "\011\011\011\011\011Some Basic Linear Algebra" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 291 29 "Draft \+ Version .8 for Maple VI" }}{PARA 0 "" 0 "" {TEXT -1 83 " \+ (Thanks to Alex Smith for much work on this.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "This worksheet documents the newer more numerically efficient Line arAlgebra package." }}{PARA 0 "" 0 "" {TEXT -1 96 "The traditional Map le linear algebra package is called linalg. The older linalg may have \+ greater" }}{PARA 0 "" 0 "" {TEXT -1 72 "symbolic capability but Linear Algebra has some serious speed advantages." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 257 107 "Load the linear alg ebra package. Below is the list of routines. You can get\nhelp on any \+ of them by typing ?" }{TEXT 258 14 "command_name " }{TEXT 259 33 "and then hitting the key." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "w ith(LinearAlgebra);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7fq%$AddG%(Adjo intG%3BackwardSubstituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/Bidiago nalFormG%-BilinearFormG%5CharacteristicMatrixG%9CharacteristicPolynomi alG%'ColumnG%0ColumnDimensionG%0ColumnOperationG%,ColumnSpaceG%0Compan ionMatrixG%0ConditionNumberG%/ConstantMatrixG%/ConstantVectorG%2Create PermutationG%-CrossProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%/ DiagonalMatrixG%*DimensionG%+DimensionsG%+DotProductG%,EigenvaluesG%-E igenvectorsG%&EqualG%2ForwardSubstituteG%.FrobeniusFormG%2GenerateEqua tionsG%/GenerateMatrixG%2GetResultDataTypeG%/GetResultShapeG%5GivensRo tationMatrixG%,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3HermitianTra nsposeG%/HessenbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/Identity MatrixG%2IntersectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimilarG%*I sUnitaryG%2JordanBlockMatrixG%+JordanFormG%(LA_MainG%0LUDecompositionG %-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%.MatrixInverseG %5MatrixMatrixMultiplyG%+MatrixNormG%5MatrixScalarMultiplyG%5MatrixVec torMultiplyG%2MinimalPolynomialG%&MinorG%)MultiplyG%,NoUserValueG%%Nor mG%*NormalizeG%*NullSpaceG%3OuterProductMatrixG%*PermanentG%&PivotG%0Q RDecompositionG%-RandomMatrixG%-RandomVectorG%%RankG%$RowG%-RowDimensi onG%-RowOperationG%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-ScalarV ectorG%*SchurFormG%/SingularValuesG%*SmithFormG%*SubMatrixG%*SubVector G%)SumBasisG%0SylvesterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG%0T ridiagonalFormG%+UnitVectorG%2VandermondeMatrixG%*VectorAddG%,VectorAn gleG%5VectorMatrixMultiplyG%+VectorNormG%5VectorScalarMultiplyG%+ZeroM atrixG%+ZeroVectorG%$ZipG" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Ent ering Matrices and Vectors" }}{EXCHG {PARA 256 "" 0 "" {TEXT 260 22 "E nter a 2 by 3 matrix:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A1 := Matr ix(2,3,[[1,2,3],[4,5,6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-% 'RTABLEG6$\"*WM!)Q\"-%'MATRIXG6#7$7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"' " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 9 "Or Simply" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "A1 := Matrix([[1,2,3],[4,5,6]]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#A1G-%'RTABLEG6$\"*s^!)Q\"-%'MATRIXG6#7$7%\"\"\"\" \"#\"\"$7%\"\"%\"\"&\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 29 "Vec tors are entered similarly" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } {TEXT 295 0 "" }{MPLTEXT 1 0 43 "v1 := Vector([5,3,1]); v2 := Vector([ 3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G-%'RTABLEG6$\"*?owQ\"- %'MATRIXG6#7%7#\"\"&7#\"\"$7#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#v2G-%'RTABLEG6$\"*gowQ\"-%'MATRIXG6#7$7#\"\"$7#\"\"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "Matrix Multiplication, Evalm, Transpose, \+ Determinant, Inverses" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" } {TEXT 261 42 "'.' is the matrix multiplication operator." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(A1 . v1) + v2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*SpwQ\"-%'MATRIXG6#7$7#\"#<7#\"#X" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A2 := A1 . Transpose(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'RTABLEG6$\"*7w\")Q\"-%'MATRIXG6#7$7 $\"#9\"#K7$F/\"#x" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 262 53 "Ordinary \+ scalar multiplication is indicated with a *." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "2*A2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$ \"*[&>)Q\"-%'MATRIXG6#7$7$\"#G\"#k7$F-\"$a\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Determinant(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#a" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 263 65 "Calculate the in verse of a matrix. \nA2^(-1) would also work here." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "A2_inv := MatrixInverse(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'A2_invG-%'RTABLEG6$\"*c:#)Q\"-%'MATRIXG6#7$7$#\"#x\" #a#!#;\"#F7$F1#\"\"(F3" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 "The M ap Function - Applying a function to every entry of a matrix." }} {EXCHG {PARA 256 "" 0 "" {TEXT 264 26 "Convert to floating point." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "map(evalf, A2_inv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*#pB)Q\"-%'MATRIXG6#7$7$$\"+Ef#fU\"! \"*$!+Ef#f#f!#57$F/$\"+$f#f#f#F1" }}}}{EXCHG {PARA 259 "" 0 "" {TEXT 265 84 "See the worksheet \"Entering Matrices\" for more information o n constructing matrices." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 48 "Base s, Rank and Subspaces Associated to a Matrix" }}{EXCHG {PARA 256 "" 0 "" {TEXT 292 37 "\nCompute a basis for the range of A1." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "ColumnSpace(Transpose(A1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$-%'RTABLEG6$\"*SrwQ\"-%'MATRIXG6#7%7#\"\"\"7#\"\"!7 #!\"\"-F%6$\"*!=n(Q\"-F)6#7%F.F,7#\"\"#" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 266 1 " " }}{PARA 256 "" 0 "" {TEXT 267 21 "Basis for the kernel ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "NullSpace(Matrix(3,3,[[1,2,3], [2,4,6],[3,6,9]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'RTABLEG6$ \"*gswQ\"-%'MATRIXG6#7%7#!\"#7#\"\"\"7#\"\"!-F%6$\"*StwQ\"-F)6#7%7#!\" $F0F." }}}{EXCHG {PARA 256 "" 0 "" {TEXT 268 1 " " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "Rank(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 269 58 "Find a basis for the subspac e spanned by a set of vectors." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "B asis(\{Vector([1,0,1]),Vector([1,0,5]),Vector([3,0,1])\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'RTABLEG6$\"*?uwQ\"-%'MATRIXG6#7%7#\"\" \"7#\"\"!F,-F%6$\"*+vwQ\"-F)6#7%7#\"\"$F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "JordanForm(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'RTABLEG6$\"*O7%)Q\"-%'MATRIXG6#7$7$,&#\"#\"*\"\"#\"\"\"*&#F0F/F0-% %sqrtG6#\"%l!)F0F0\"\"!7$F7,&F-F0*&#F0F/F0*$F3F0F0!\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Solution of a Linear System" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "LinearSolve(Matrix([[a,b],[c,d]]),V ector([e,f]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*SvwQ \"-%'MATRIXG6#7$7#,$*&,&*&%\"bG\"\"\"%\"fGF1F1*&%\"eGF1%\"dGF1!\"\"F1, &*&%\"aGF1F5F1F1*&%\"cGF1F0F1F6F6F67#*&,&*&F9F1F2F1F1*&F;F1F4F1F6F1F7F 6" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Dot Product and Cross Prod uct" }}{EXCHG {PARA 256 "" 0 "" {TEXT 293 12 "Dot product." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "DotProduct(v1, Vector([1,4,8]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 270 21 "Cross product in R^3." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "CrossP roduct(v1,Vector([1,0,0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTAB LEG6$\"*?wwQ\"-%'MATRIXG6#7%7#\"\"!7#\"\"\"7#!\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Rows, Columns, and Submatrices" }}{EXCHG {PARA 256 "" 0 "" {TEXT 271 49 "Copy the all the entries of A1 into the matr ix B." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B:= Matrix(3,3, A1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\"*g'\\)Q\"-%'MATRIX G6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"!F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 2 "Or" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "B:= Ma trix(3,3); B[2..3,1..3] := A1; B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"BG-%'RTABLEG6$\"*S.&)Q\"-%'MATRIXG6#7%7%\"\"!F.F.F-F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"BG6$;\"\"#\"\"$;\"\"\"F)-%'RTABLEG6$\"*s^!) Q\"-%'MATRIXG6#7$7%F+F(F)7%\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*S.&)Q\"-%'MATRIXG6#7%7%\"\"!F,F,7%\"\"\" \"\"#\"\"$7%\"\"%\"\"&\"\"'" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 272 12 "Row 1 of A1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Row(A1,1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*SxwQ\"-%'VECTORG6#7%\" \"\"\"\"#\"\"$" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 273 15 "Column 1 of \+ A1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Column(A1,1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*!yn(Q\"-%'MATRIXG6#7$7#\"\"\"7#\" \"%" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 274 56 "Construct a submatrix o f A using rows 1..2 and column 1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "SubMatrix(A1,1..2,1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA BLEG6$\"*'ze)Q\"-%'MATRIXG6#7$7#\"\"\"7#\"\"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 5 "Norms" }}{EXCHG {PARA 256 "" 0 "" {TEXT 275 105 "The \+ norm function uses absolute value signs which is somewhat inconvenient for algebraic \nsimplification." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Norm(Vector([x,y]),2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6 #,&*$)-%$absG6#%\"xG\"\"#\"\"\"F/*$)-F+6#%\"yGF.F/F/F/" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 276 61 "So it may be better top define your ow n vector norm function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "vnorm := u -> sqrt(DotProduct(u,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&vno rmGR6#%\"uG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#-_%.LinearAlgebraG%+Dot ProductG6$9$F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "vno rm(Vector([x,y]),2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,& *&-%*conjugateG6#%\"xG\"\"\"F,F-F-*&-F*6#%\"yGF-F1F-F-F-" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 277 37 "The default norm is the infinity one. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Norm(Vector([x,y]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$maxG6$-%$absG6#%\"xG-F'6#%\"yG" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 87 "Using Maple's Solve Command to fi nd the set of matrices which\ncommute with a given one." }}{EXCHG {PARA 256 "" 0 "" {TEXT 278 198 "Matrix equations can be converted int o sets of equations and then solved. \nFor example consider the proble m of finding all matrices B which commute with the matrix A2 above (i. e. A2 B - B A2 = 0) :\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "B := Mat rix(2,2,[[a,b],[c,d]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RT ABLEG6$\"*?K')Q\"-%'MATRIXG6#7$7$%\"aG%\"bG7$%\"cG%\"dG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "mat_with_unknowns := A2 . B - B . A 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2mat_with_unknownsG-%'RTABLEG6 $\"*o_')Q\"-%'MATRIXG6#7$7$,&%\"cG\"#K*&F0\"\"\"%\"bGF2!\"\",(F3!#j*&F 0F2%\"dGF2F2*&F0F2%\"aGF2F47$,(F:F0*&\"#jF2F/F2F2*&F0F2F8F2F4,&F3F0*&F 0F2F/F2F4" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 279 316 " Now we solve fo r the unknown entries a,b,c, and d.\n(This relies on the fact that Map le's solve command ordinarily wants a set of equations in its\nfirst a rgument. But if given a first argument which is a set of algebraic equ ations,\nit interprets these as a set of equations by viewing each exp ression as equal to 0.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "soln1 := solve(convert(mat_with_unknowns,set),\{a,b,c,d\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&soln1G<&/%\"dGF'/%\"cGF)/%\"aG,&F)#!#j\"#KF'\"\"\" /%\"bGF)" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 280 68 "This globally assi gns b and a to have the values mentioned in soln1." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "assign(soln1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLE G6$\"*?K')Q\"-%'MATRIXG6#7$7$,&%\"cG#!#j\"#K%\"dG\"\"\"F-7$F-F1" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Gram Schmidt, GenerateEquations, \+ LinearSolve" }}{EXCHG {PARA 256 "" 0 "" {TEXT 281 129 "Convert a list \+ of vectors into a list of orthogonal vectors. The first k vectors in\n each list (for any k) span the same subspace." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "GramSchmidt([Vector([1,1,1]),Vector([1,2,3]),Vector([ 1,4,5])]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%'RTABLEG6$\"*SzwQ\"- %'MATRIXG6#7%7#\"\"\"F,F,-F%6$\"*?!o(Q\"-F)6#7%7#!\"\"7#\"\"!F,-F%6$\" *+\"o(Q\"-F)6#7%7##F5\"\"$7##\"\"#F@F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 92 "GenerateEquations can be used to convert a matrix into l inear equations in a number of ways:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "GenerateEquations(A1, ['x','y']);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,&%\"xG\"\"\"*&\"\"#F'%\"yGF'F'\"\"$/,&F&\"\"%*&\"\"&F'F*F'F' \"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "GenerateEquations( A1, ['x','y','z']);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,(%\"xG\"\" \"*&\"\"#F'%\"yGF'F'*&\"\"$F'%\"zGF'F'\"\"!/,(F&\"\"%*&\"\"&F'F*F'F'*& \"\"'F'F-F'F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Generate Equations(A1, ['x','y','z'], <7,13>);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,(%\"xG\"\"\"*&\"\"#F'%\"yGF'F'*&\"\"$F'%\"zGF'F'\"\"(/,(F&\"\"% *&\"\"&F'F*F'F'*&\"\"'F'F-F'F'\"#8" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 45 "LinearSolve solves the equation A.x = b for x" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "LinearSolve(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*!=o(Q\"-%'MATRIXG6#7$7#!\"\"7#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "LinearSolve(A2, <3, 16>);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*g#o(Q\"-%'MATRIXG6#7$7##!$\"G \"#a7##\"#k\"#F" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Eigenvalues \+ and Eigenvectors" }}{EXCHG {PARA 256 "" 0 "" {TEXT 282 10 "A list of \+ " }{TEXT 283 8 "symbolic" }{TEXT 284 14 " eigenvalues." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "evals_of_A2 := Eigenvalues(A2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%,evals_of_A2G-%'RTABLEG6$\"*#4kp8-%'MATRIXG6#7 $7#,&#\"#\"*\"\"#\"\"\"*&#F2F1F2-%%sqrtG6#\"%l!)F2F27#,&F/F2*&#F2F1F2* $F5F2F2!\"\"" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 15 "evals_of_ A2[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"#\"*\"\"#\"\"\"*&#F'F&F '-%%sqrtG6#\"%l!)F'F'" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 285 43 "Extra ct the first eigenvalue from the list." }}{PARA 256 "" 0 "" {TEXT 286 56 "Convert the first eigenvalue to a floating point number." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(evals_of_A2[1]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+_sES!*!\")" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 287 71 "Convert evals_of_A2 to a list, and change each entry to floati ng point." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "map(evalf, evals_of_A2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*+$o(Q\"-%'MATRIXG 6#7$7#$\"+_sES!*!\")7#$\")[FtfF." }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 288 41 "Directly calculate numerical eigenvalues." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(Eigenvalues(A2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*!Qo(Q\"-%'MATRIXG6#7$7#$\"+_s ES!*!\")7#$\")[FtfF." }}}{EXCHG {PARA 256 "" 0 "" {TEXT 289 46 " Eigen vectors outputs both values and vectors." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evects_of_A2 := Eigenvectors(A2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%-evects_of_A2G6$-%'RTABLEG6$\"*g%o(Q\"-%'MATRIXG6#7 $7#,&#\"#\"*\"\"#\"\"\"*&#F3F2F3-%%sqrtG6#\"%l!)F3F37#,&F0F3*&#F3F2F3* $F6F3F3!\"\"-F'6$\"*CN*)Q\"-F+6#7$7$F3F37$,&#\"#j\"#kF3*&#F3FKF3F6F3F3 ,&FIF3*&#F3FKF3F>F3F?" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 290 39 "Check the eigenvectors and eigenvalues." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "c_poly := CharacteristicPolynomial(Transpose(A1).A1, lambda);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'c_polyG,(%'lambdaG\"#a*&\"#\"*\"\" \")F&\"\"#F*!\"\"*$)F&\"\"$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "simplify(subs('lambda' = evals_of_A2[1], c_poly));\nsimplify( subs('lambda' = evals_of_A2[2], c_poly));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "evector1 := Column(evects_of _A2[2],1); evector2 := Column(evects_of_A2[2],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)evector1G-%'RTABLEG6$\"*!eo(Q\"-%'MATRIXG6#7$7#\"\" \"7#,&#\"#j\"#kF.*&#F.F3F.-%%sqrtG6#\"%l!)F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)evector2G-%'RTABLEG6$\"*g'o(Q\"-%'MATRIXG6#7$7#\"\" \"7#,&#\"#j\"#kF.*&#F.F3F.*$-%%sqrtG6#\"%l!)F.F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 44 "Maple doesn't always simplify when it sh ould" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "(A2 . evector1) - (evals_of _A2[1] . evector1);\n(A2 . evector2) - (evals_of_A2[2] . evector2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*S(o(Q\"-%'MATRIXG6#7$7 #\"\"!7#,(#\"%**o\"#k\"\"\"*&#\"#xF1F2-%%sqrtG6#\"%l!)F2F2*&,&#\"#jF1F 2*&#F2F1F2F6F2F2F2,&#\"#\"*\"\"#F2*&#F2FCF2F6F2F2F2!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*!yo(Q\"-%'MATRIXG6#7$7#\"\"!7#,(# \"%**o\"#k\"\"\"*&#\"#xF1F2*$-%%sqrtG6#\"%l!)F2F2!\"\"*&,&#\"#jF1F2*&# F2F1F2F6F2F;F2,&#\"#\"*\"\"#F2*&#F2FEF2F6F2F;F2F;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 123 "map(simplify, (A2 . evector1) - (evals_of_A 2[1] . evector1));\nmap(simplify, (A2 . evector2) - (evals_of_A2[2] . \+ evector2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*g)o(Q\"- %'MATRIXG6#7$7#\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$ \"*S*o(Q\"-%'MATRIXG6#7$7#\"\"!F+" }}}}}{MARK "2 2 0" 72 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 138803444 138805172 138766820 138766860 138766940 138817612 138819548 138821556 138823692 138767140 138767180 138767260 138767340 138767420 138767500 138841236 138767540 138767620 138849660 138850340 138767740 138767780 138858796 138863220 138865268 138767940 138768020 138768100 138768180 138768260 136964092 138768300 138768380 138768460 138893524 138768580 138768660 138768740 138768780 138768860 138768940 }{RTABLE M6R0 I6RTABLE_SAVE/138803444X,%)anythingG6"6"][[[[[p'"#"$"""""%""#""&""$""'6" } {RTABLE M6R0 I6RTABLE_SAVE/138805172X,%)anythingG6"6"][[[[[p'"#"$"""""%""#""&""$""'6" } {RTABLE M6R0 I6RTABLE_SAVE/138766820X*%)anythingG6"6"\[[[[[t$"$""&""$"""6" } {RTABLE M6R0 I6RTABLE_SAVE/138766860X*%)anythingG6"6"\[[[[[t#"#""$""%6" } {RTABLE M6R0 I6RTABLE_SAVE/138766940X*%)anythingG6"6"\[[[[[t#"#"#<"#X6" } {RTABLE M6R0 I6RTABLE_SAVE/138817612X,%)anythingG6"6"][[[[[p%"#"#"#9"#KF("#x6" } {RTABLE M6R0 I6RTABLE_SAVE/138819548X,%)anythingG6"6"][[[[[p%"#"#"#G"#kF("$a"6" } {RTABLE M6R0 I6RTABLE_SAVE/138821556X,%)anythingG6"6"][[[[[p%"#"##"#x"#a#!#;"#FF*#""(F,6" } {RTABLE M6R0 I6RTABLE_SAVE/138823692X,%)anythingG6"6"][[[[[p%"#"#$"+Ef#fU"!"*$!+Ef#f#f!#5F*$ "+$f#f#f#F,6" } {RTABLE M6R0 I6RTABLE_SAVE/138767140X*%)anythingG6"6"\[[[[[t$"$"""""!!""6" } {RTABLE M6R0 I6RTABLE_SAVE/138767180X*%)anythingG6"6"\[[[[[t$"$""!"""""#6" } {RTABLE M6R0 I6RTABLE_SAVE/138767260X*%)anythingG6"6"\[[[[[t$"$!"#"""""!6" } {RTABLE M6R0 I6RTABLE_SAVE/138767340X*%)anythingG6"6"\[[[[[t$"$!"$""!"""6" } {RTABLE M6R0 I6RTABLE_SAVE/138767420X*%)anythingG6"6"\[[[[[t$"$"""""!F'6" } {RTABLE M6R0 I6RTABLE_SAVE/138767500X*%)anythingG6"6"\[[[[[t$"$""$""!"""6" } {RTABLE M6R0 I6RTABLE_SAVE/138841236X,%)anythingG6"6"][[[[[p%"#"#,&#"#"*""#"""*$"%l!)#F+F*F. 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