MATH225 : Notes Done in LaTeX
22 Dec 2011
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This means that the entries on the opposite sides of the main diagonal of a are mutally equal. This is true only if m = n. so any symmetric matrix is a square matrix. If a is symmetric, then any two eigenvectors from di erent eigenspaces are orthogonal. Let u, v be two eigenvectors corresponding to distinct eigenvalues 1, 2. We should show that u v = 0. 1 (u v) = ( 1u) v. 1 (u v) = (cid:0)ut at(cid:1) v. 1 (u v) = (cid:0)ut a(cid:1) v (cid:0)ut v(cid:1) 1 (u v) = 2 (u v) = . 0 = 2 (u v) 1 (u v) 0 = ( 2 1) (u v) u1 u2 = 0. If the symmetric matrix a is diagonalizable, then (cid:3) A = p dp 1 for an invertiable matrix p and a diagonal matrix d. the columns of p are eigenvectors of a.
