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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
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
We have the expression λI − A = λ−3
λ+2 ; so that A has the characteristic polynomial
pA (λ) = det
= (λ − 3) (λ + 2) + 4 · 1 = λ2 − λ − 2 = (λ − 2) (λ + 1) .
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.
- Spring '09