MATH225 : Eigenvalues and Eigenvectors
22 Dec 2011
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1. 5. 2 changing basis to make a matrix diagonalizable . An eigenvector of an n n matrix a is a nonzero vector x such that for some scalar . A (possibly zero) scalar is called an eigenvalue of a if there is a nonzero vector x such that ax = x; such an x is called an eigenvector corresponded to . We need to see if there is any x (cid:54)= 0 such that. Ax ix = 0 (a i) x = 0. So is an eigenvalue of a if (a i) x = 0 has a nontrivial solution. (a i) x = (cid:20) 3 2. Coe cient matrix for (a i) x = 0 (cid:20) 1 2 | 0. 0 0 | 0 (cid:21) = (cid:20) 1 2 | 0 (cid:40)x1 + 2x2 = 0 (cid:40)x1 = 2x2 x2 = x2. Solution: (a i) x = 0 = (cid:26)x2(cid:20) 2. Ax = x (cid:20) 2x2 x2 (cid:21) = x.
