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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[r%$AddG%(AdjointG%3BackwardSubst
ituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-BilinearF
ormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG%0Colum
nDimensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMatrixG%0Conditi
onNumberG%/ConstantMatrixG%/ConstantVectorG%2CreatePermutationG%-Cross
ProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%/DiagonalMatrixG%*Di
mensionG%+DimensionsG%+DotProductG%6EigenConditionNumbersG%,Eigenvalue
sG%-EigenvectorsG%&EqualG%2ForwardSubstituteG%.FrobeniusFormG%4Gaussia
nEliminationG%2GenerateEquationsG%/GenerateMatrixG%2GetResultDataTypeG
%/GetResultShapeG%5GivensRotationMatrixG%,GramSchmidtG%-HankelMatrixG%
,HermiteFormG%3HermitianTransposeG%/HessenbergFormG%.HilbertMatrixG%2H
ouseholderMatrixG%/IdentityMatrixG%2IntersectionBasisG%+IsDefiniteG%-I
sOrthogonalG%*IsSimilarG%*IsUnitaryG%2JordanBlockMatrixG%+JordanFormG%
(LA_MainG%0LUDecompositionG%-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%
*MatrixAddG%.MatrixInverseG%5MatrixMatrixMultiplyG%+MatrixNormG%5Matri
xScalarMultiplyG%5MatrixVectorMultiplyG%2MinimalPolynomialG%&MinorG%(M
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omMatrixG%-RandomVectorG%%RankG%6ReducedRowEchelonFormG%$RowG%-RowDime
nsionG%-RowOperationG%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-Scal
arVectorG%*SchurFormG%/SingularValuesG%*SmithFormG%*SubMatrixG%*SubVec
torG%)SumBasisG%0SylvesterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG
%0TridiagonalFormG%+UnitVectorG%2VandermondeMatrixG%*VectorAddG%,Vecto
rAngleG%5VectorMatrixMultiplyG%+VectorNormG%5VectorScalarMultiplyG%+Ze
roMatrixG%+ZeroVectorG%$ZipG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "A := Matrix([[2,4,3],[-4,-6,-3],[3,3,1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")#>!*G\"-%'MATRIXG6#7%7%\"\"#\"\"%
\"\"$7%!\"%!\"'!\"$7%F0F0\"\"\"%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "CharacteristicPolynomial(A, x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"F(*&F'F()F&\"\"#F(F(\"\"%!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p:=CharacteristicPolynomial
(A, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*$)%\"xG\"\"$\"\"\"
F**&F)F*)F(\"\"#F*F*\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "solve(p(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%f*6#%\"xG6
\"F&F&\"\"\"F&F&F&f*F$F&F&F&!\"#F&F&F&f*F$F&F&F&F)F&F&F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 112 "The eigevalues are 1 and -2. Next load t
he \"linalg\" package. This is different from the \"LinearAlgebra\" pa
ckage." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 104 "Warning, the previous binding of the nam
e GramSchmidt has been removed and it now has an assigned value\n" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 144 "Next define the 3 by 3 identity matrix Id (in Maple, \"I
\" is reserved for the imaginary number i). Also define the three dime
nsional zero vector." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Id:
=Matrix(3,3,shape=identity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG
-%'RTABLEG6%\")g++9-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/F/F.%'Matr
ixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "zero := Vector[colum
n]([0,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%zeroG-%'RTABLEG6%\"
)#>7^\"-%'MATRIXG6#7%7#\"\"!F-F-&%'VectorG6#%'columnG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 157 "It seems that 1 and -2 are the eigenvalu
es, since 1 is a root and -2 is a double root of the characteristic po
lynomial. What are the associated eigenvectors?" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "First find the eigen
vector for the eigenvalue 1. Solve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 7 "Av = 1v" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 24 "by using linalg to solve" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "(A-1I)v = 0." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The zero \+
on the right  of the above equationis the zero vector!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "linsolve(A-Id, zero);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%&%#_tG6#\"\"\",$F'!\"\"F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "Next get the eigenvector for the \+
eigenvalue -2. It turns out there is only one eigenvector even though \+
-2 is a double root of the characteristic polynomial. What would you d
o if there were multiple eigenvectors for the same eigenvalue?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "linsolve(A+2.Id, zero);" }}
{PARA 8 "" 1 "" {TEXT -1 31 "Error, missing operator or `;`\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Does the above attempt work? If no
t, try again a different way:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "linsolve(A+ScalarMultiply(Id,2), zero);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%,$&%#_tG6#\"\"\"!\"\"F(\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 78 "What is a faster way to find the eigenvaluesand their associate
d eigenvectors?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Eigenvec
tors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")#>aS\"-%'MA
TRIXG6#7%7#\"\"\"7#!\"#F-&%'VectorG6#%'columnG-F$6%\")gJ=9-F(6#7%7%!\"
\"F:\"\"!7%F,F,F;7%F:F;F;%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 168 "The entries of the outputted vector are the eigenvalues. The c
olumns of the outputted matrix are the eigenvectors (caution- the zero
 vector can not be an eigenvector!)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 112 "Sometimes an eigenvalue is a complex number with nonzero imagi
nary part (similarly for entries of eigenvectors)." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 28 "B := Matrix([[0,-1],[1,0]]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\")wP=9-%'MATRIXG6#7$7$\"\"!!
\"\"7$\"\"\"F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
Eigenvectors(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")7V0
9-%'MATRIXG6#7$7#^#\"\"\"7#^#!\"\"&%'VectorG6#%'columnG-F$6%\")/.>9-F(
6#7$7$F-F-7$F/F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Next
 we enter the eigenvectors and name them e1 and e2." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "e1 := Vector[column]([1,-I]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#e1G-%'RTABLEG6%\")G63;-%'MATRIXG6#7$7#\"
\"\"7#^#!\"\"&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "e2 := Vector[column]([1,I]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#e2G-%'RTABLEG6%\")o63;-%'MATRIXG6#7$7#\"\"\"7#^#F.&%
'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Next we ve
rify that B*e1=I*e1 (since I is the eigenvalue corresponding to e1)." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B.e1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%\".G6$%\"BG-%'RTABLEG6%\")G63;-%'MATRIXG6#7$7#\"\"\"7
#^#!\"\"&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "I.e1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")#*4;;-%'M
ATRIXG6#7$7#^#\"\"\"7#F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 72 "Verify that B*e2=-I*e2 (since -I is the eigenvalue cor
responding to e2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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