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12.38 An important theorem in linear algebra, the Cayley-Hamilton theorem, says that every square matrix satisfies its characteristic equation. Verify the Cayley-Hamilton theorem for
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1 2 3 1 A= 2 5 1 2
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1 2 1 2 3 1 3 1 a= ; 2 5 7 1 1 2 3 6 CharacteristicPolynomial[a, x] 196 + 161 x + 15 x2 13 x3 + x4 196 IdentityMatrix[4]+ 161 a + 15 MatrixPower[a, 2] 13 MatrixPower[a, 3]+ MatrixPower[a, 4] // MatrixForm 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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12.39 Approximate the eigenvalues of the 10 10 Hilbert matrix: hi j =
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hilbert = HilbertMatrix[10]; hilbert // MatrixForm
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1 1 2 1 1 2 3 1 1 3 4 1 1 4 5 1 1 5 6 1 1 6 7 1 1 7 8 1 1 8 9 1 1 9 10 1 1 10 11
1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12
1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13
1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14
1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 15
1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 15 1 16
1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 15 1 16 1 17
1 9 1 10 1 11 1 12 1 13 1 14 1 15 1 16 1 17 1 18
1 10 1 11 1 12 1 13 1 14 1 15 1 16 1 17 1 18 1 19
Eigenvalues[N[hilbert]]
{1.75192,0.34293,0.0357418,0.00253089, 0.00012875,4.72969 10 6,1.22897 10 7, 2.14744 10 9,2.26675 10 11,1.09287 10 13}
Linear Algebra
12.40 Approximate the eigenvalues of the 10 10 matrix A such that ai , j =
SOLUTION
i + j 1 if i + j 11 21 i j if i + j > 11
f[i_ , j_ ] i + j 1 /; i + j 11 f[i _, j_ ] 21 i j /; i + j > 11 a = Array[f, {10, 10}]; a // MatrixForm 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 9 3 4 5 6 7 8 9 10 9 8 4 5 6 7 8 9 10 9 8 7 5 6 7 8 9 10 9 8 7 6 6 7 8 9 10 9 8 7 6 5 7 8 9 10 9 8 7 6 5 4 8 9 10 9 8 7 6 5 4 3 9 10 9 8 7 6 5 4 3 2 10 9 8 7 6 5 4 3 2 1 Eigenvalues[N[a]] {67.8404, 20.4317, 4.45599, 2.42592, 1.39587, 1., 0.756101, 0.629808, 0.55164, 0.512543}
12.7 Diagonalization and Jordan Canonical Form
Given an n n matrix, A, if there exists an invertible matrix, P, such that A = PDP 1, where D is a diagonal matrix, we say that A is diagonalizable. Not every matrix is diagonalizable. However, it can be shown that if A has a set of n linearly independent eigenvectors, then A is diagonalizable. P is the matrix whose columns are the eigenvectors of A, and D is the diagonal matrix whose main diagonal entries are their respective eigenvalues.
EXAMPLE 30
18 8 a= 15 15
51 24 48 47
27 15 14 8 ; 28 15 25 12
Since the eigenvalues are distinct, the corresponding eigenvectors will be linearly independent.
Eigenvalues[a] {4, 3, 2, 1} Eigenvectors[a] {{3, 1, 2, 3},{1, 0, 0, 1}, {0, 1, 3, 2},{3, 2, 3, 2}} p = Transpose[Eigenvectors[a]] {{3, 1, 0, 3}, {1, 0, 1, 2}, {2, 0, 3, 3}, {3, 1, 2, 2}} d = DiagonalMatrix[Eigenvalues[a]]
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