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lecture_31 (dragged) 2 - MA 36600 LECTURE NOTES: FRIDAY,...

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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, APRIL 10 3 Characteristic Polynomials. We give a simple way to compute eigenvalues of an n × n matrix A. Consider the equation Ax = λx =⇒ (λ I − A) x = 0. Since this holds for a nonzero vector x, we see that the matrix λ I − A must be singular. We found before that an n × n matrix is singular if and only if its determinant is zero. Hence we conclude that λ is an eigenvalue for A if and only if λ is a root of the characteristic polynomial pA (λ) = det (λ I − A) . Example. We compute the eigenvalues of the 2 × 2 matrix ￿ ￿ 3 −1 A= . 4 −2 We have the expression λI − A = ￿ λ−3 −4 1 λ+2 ￿ ; so that A has the characteristic polynomial ￿ ￿ λ−3 1 pA (λ) = det = (λ − 3) (λ + 2) + 4 · 1 = λ2 − λ − 2 = (λ − 2) (λ + 1) . −4 λ+2 Hence the eigenvalues of A are λ1 = 2 and λ2 = −1. ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

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