symmetric_matrices_2009_10_05_02

symmetric_matrices_2009_10_05_02 - 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 C . Then Ax = x and x negationslash = 0 we will show that = . we know that x T Ax = x T x = n summationdisplay i =1 | x i | 2 also x T Ax = ( Ax ) T x = x T x = n summationdisplay i =1 | x i | 2 since bardbl x bardbl negationslash = 0 , we have = 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 = 1 x 1 and Ax 2 = 2 x 2 . Then x T 1 Ax 2 = ( Ax 1 ) T x 2 = 1 x T 1 x 2 and also x T 1 Ax 2 = 2 x T 1 x 2 therefore ( 1 2 ) x T 1 x 2 = 0 and since 1 negationslash = 2 we must have x T 1 x 2 = 0 if A is symmetric and has n distinct eigenvalues, then its eigenvectors form an or- thogonal basis for the vector space R n 6 - 4...
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symmetric_matrices_2009_10_05_02 - 6 - 1 Symmetric and...

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