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symmetric_matrices_2009_10_05_02_2up

symmetric_matrices_2009_10_05_02_2up - 6 1 Symmetric and...

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Unformatted text preview: 6 - 1 Symmetric and Positive Matrices S. Lall, Stanford 2009.10.05.02 6. Symmetric and Positive Matrices • Eigenvalues • Eigenvectors • Example: mechanical systems • Quadratic forms • Positive-definite matrices • Matrix square-roots • Example: mechanical systems • Ellipsoids • Example: navigation 6 - 2 Symmetric and Positive Matrices S. Lall, Stanford 2009.10.05.02 Eigenvalues of Symmetric Matrices a matrix A ∈ R n × n is called symmetric if A = A T eigenvalues if A is symmetric, then the eigenvalues of A are real to show this, suppose x is an eigenvector of A with eigenvalue λ ∈ C . Then Ax = λx and x 6 = 0 we will show that λ = ¯ λ . we know that ¯ x T Ax = λ ¯ x T x = λ n X i =1 | x i | 2 also ¯ x T Ax = ( Ax ) T x = ¯ λ ¯ x T x = ¯ λ n X i =1 | x i | 2 since k x k 6 = 0 , we have λ = ¯ λ 6 - 3 Symmetric and Positive Matrices S. Lall, Stanford 2009.10.05.02 Eigenvectors of Symmetric Matrices for symmetric matrices: eigenvectors corresponding to distinct eigenvalues are orthogonal suppose Ax 1 = λ 1 x 1 and Ax 2 = λ 2 x 2 . Then x T 1 Ax 2 = ( Ax 1 ) T x 2 = λ 1 x T 1 x 2 and also x T 1 Ax 2 = λ 2 x T 1 x 2 therefore ( λ 1 − λ 2 ) x T 1 x 2 = 0 and since λ 1 6 = λ 2 we must have x T 1 x 2 = 0 • if A is symmetric and has n distinct eigenvalues, then its eigenvectors form an...
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symmetric_matrices_2009_10_05_02_2up - 6 1 Symmetric and...

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