MATA23H3 Lecture Notes - Lecture 21: Eigenvalues And Eigenvectors, Linear Map, Automobilclub Von Deutschland
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Mata23 - lecture 21 - characteristic polynomials and properties of eigenvalues. Characteristic polynomials: let a be an n n matrix. The characteristic polynomial of a is given by p( ) = det(a i). It contains the zero vector and all the eigenvectors of a corresponding to . Remark: since p( ) = det(a i) has at most n distinct roots, a n n matrix a has at most n distinct eigenvalues (cid:20) 3. 5: example 2: find the eigenvalues and corresponding eigenspace of a = Set p( ) = 0 = 2 1 = 0. 1 = 1, 2 = 1 are eigenvalues of a. For 1 = 1, (a 1i)(cid:126)x = (a i)(cid:126)x = (cid:126)0. A i = (cid:20)1 0 (cid:21) (cid:21) Let y = t = t = 2x. Set y t t (cid:34) 1 (cid:26) 1. A 2i = a + i = (cid:20) 3. (a 2i)(cid:126)x = (cid:126)0 (cid:20) 2 (cid:20)x.
