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One basis for P 5 is the set v = {1, x, x 2 , x 3 , x 4 , x 5 }. They comprise a linearly independent set that spans P 5 . v = {1, x, x2, x3, x4, x5}; f[p_, q_]
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Orthogonalize[v, f] //Simplify
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{1, 3( 1+2x) 5(1 6x +6x2) 7( 1+12x 30x2 +20x3) , , , 3(1 20x +90x2 140x3 +70x4) 11( 1+30x 210x2 + 560x3 630x4 +252x5)} ,
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12.6 Eigenvalues and Eigenvectors
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is said to be an eigenvalue of a square matrix, A, if there exists a non-zero vector, x, such that Ax = x. As powerful as the eigenvalue concept is in linear algebra, the computation of eigenvalues and their corresponding eigenvectors can be extremely difficult if the matrix is large. One way to determine the eigenvalues of a matrix is to solve the characteristic equation det(A I) = 0. Once the eigenvalues are determined, the eigenvectors can be found by solving a homogeneous linear system.
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4 1 1 a = 2 5 2 ; 1 2 2 length = Length[a]; Solve[Det[a k IdentityMatrix[length]] 0, k] {{ 3}, {{ 3}, { 5}}
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The eigenvalues are 3 (with multiplicity 2) and 5. To find the eigenvectors, we look at the null space of A I:
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NullSpace[a 3 IdentityMatrix[length]] {{1, 0, 1}, { 1, 1, 0}} NullSpace[a 5 IdentityMatrix[length]] {{1, 2, 1}}
Linear Algebra
Of course, as one might expect, Mathematica contains commands that automatically compute eigenvalues, eigenvectors, and some other related items.
CharacteristicPolynomial[matrix, var] returns the characteristic polynomial of matrix expressed in terms of variable var. Eigenvalues[matrix] returns a list of the eigenvalues of matrix. Eigenvectors[matrix] returns a list of the eigenvectors of matrix. Eigensystem[matrix] returns a list of the form {eigenvalues, eigenvectors}.
EXAMPLE 28
4 1 1 a = 2 5 2 ; 1 2 2 CharacteristicPolynomial[a, x] 45 39 x + 11 x2 x3 Eigenvalues[a] {5, 3, 3} Eigenvectors[a]
{{1, 2, 1}, {1, 0, 1}, { 1, 1, 0}}
Eigensystem[a]
{{5, 3, 3}, {{1, 2, 1},{1, 0, 1}, { 1, 1, 0}}}
If the entries of the matrix are expressed exactly, i.e., in non-decimal form, Mathematica tries to determine the eigenvalues and eigenvectors exactly. If any of the entries of the matrix are expressed in decimal form, Mathematica returns decimal approximations. Alternatively, one can use N[matrix] as the argument of CharacteristicPolynomial, Eigenvalues, Eigenvectors, and Eigensystem to force Mathematica to return decimal eigenvalues and eigenvectors. If k digit precision is desired, N[matrix, k] will return k significant digits.
EXAMPLE 29
1 1 a= ; 1 3 Eigenvalues[a]
{2 +
2,2 2
} }}
Eigenvectors[a]
{{ 1 +
2, 1 , 1 2, 1
Eigenvalues[N[a]] {3.41421, 0.585786} Eigenvectors[N[a]] {{0.382683, 0.92388}, { 0.92388, 0.382683}} Eigenvalues[N[a, 20]] {3.4142135623730950488, 0.58578643762690495120} Eigenvectors[N[a, 20]] {{ 0.38268343236508977173, 0.92387953251128675613}, { 0.92387953251128675613, 0.38268343236508977173}} Eigensystem[N[a]] {{3.41421, 0.585786}, {{0.382683, 0.92388}, { 0.92388, 0.382683}}}
Linear Algebra
Note: Because different algorithms are used for computing numerical eigenvalues, they sometimes emerge in a different order. Furthermore, since eigenvectors are not uniquely determined, the numerical eigenvectors may appear to be multiples or linear combinations of those obtained previously.
SOLVED PROBLEMS
12.36 What is the characteristic polynomial of the matrix A, whose entries are the first 25 consecutive integers
SOLUTION
a = Partition[Range[25], 5]; a // MatrixForm 1 2 3 4 6 7 8 9 11 12 13 14 16 17 18 19 21 22 23 24 5 10 15 20 25
CharacteristicPolynomial[a, x] 250 x3 + 65 x4 x5
12.37 Consider the tridiagonal 5 5 matrix whose main diagonal entries are 4, with 1s on the adjacent diagonals. Show the eigenvalues and corresponding eigenvectors in a clear, unambiguous manner.
SOLUTION
m = Table[If[Abs[i j] 1, 1, If[i j, 4, 0]], {i, 1, 5}, {j, 1, 5}]; m // MatrixForm 4 1 0 0 0 1 4 1 0 0 0 1 4 1 0 0 0 1 4 1 0 0 0 1 4
data = Eigensystem[m] {{4+ 3, 5, 4, 3, 4, 3},{{1, 3, 2, 3, 1},{ 1, 1, 0, 1, 1}, } {1, 0, 1, 0, 1},{ 1, 1, 0, 1, 1},{1, 3, 2, 3, 1}}} } Do[Print["eigenvalue #", k, "is", data[[1, k]], "with corresponding eigenvector:", data[[2, k]]], {k, 1, 5}] eigenvalue #1 is 4 +
3 with corresponding eigenvector: {1, 3 , 2, 3 , 1}
eigenvalue #2 is 5 with corresponding eigenvector: { 1, 1, 0, 1, 1} eigenvalue #3 is 4 with corresponding eigenvector: {1, 0, 1, 0, 1} eigenvalue #4 is 3 with corresponding eigenvector: { 1, 1, 0, 1, 1} eigenvalue #5 is 4
3 with corresponding eigenvector: {1, 3 , 2, 3 , 1}
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