Orthogonal Diagonalization A Matrix A Is Orthogonally Diagonalizable If There Is An Orthogonal Matrix P With P 1 Pt 1 (80.11 KiB) Viewed 27 times
Orthogonal diagonalization: A matrix A is orthogonally diagonalizable if there is an orthogonal matrix P (with P-1 = PT ) and a diagonal matrix D such that A PDPT = PDP-1. = Theorem 1: An n x n matrix A is orthogonally diagonalizable if and only if A is a symmetric matrix i.e. AT = A. = 3 1 Problem 1: Let A = - 1 3 1) Is A is symmetric? Let 11, 12 be the eigenvalues of A with 11 > 12. ) 2) 21 = 3) 12 = = Let V1 = X1 Yi be the unit eigenvectors X2 corresponding to 1, and 12 respectively. Then [v2 , V2 = [ ya
4) X1 = 5) x2 = 6) yı = 7) y2 = 8) Are V1 and v2 orthogonal? = = [ ] 12 To orthogonally diagonalize A, we form the matrix 21 0 P = [v1 v2], D= Note the order 0 of vectors in P and the order of eigenvalues in D. They must correspond. If we switch the eigenvalues in D we have to switch the eigenvectors in P. In the exercise above, the two eigenvectors corresponding to different eigenvalues were orthogonal.
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