184 CHAPTER 10. EIGENVALUES AND EIGENVECTORS

M is an n×n matrix, then this means that the columns of M cannot be lineraly independentsince if they were, then by Theorem 11.5.2 M−1 would exist. Thus if A−λ I fails to havean inverse as above, then the columns are not independent and so there exists a nonzero xsuch that (A−λ I)x= 0. Thus we have the following proposition.

Proposition 10.3.4 The eigenvalues of an n× n matrix are the roots of det(A−λ I) = 0.Corresponding to each of these λ is an eigenvector.

Note that if A = S−1BS, then A,B have the same characteristic polynomial, hence thesame eigenvalues. (They might have different eigenvectors and usually will.) To see this,note that from the properties of determinants

det(A−λ I) = det(S−1BS−λS−1IS

)= det

(S−1 (B−λ I)S

)= det

(S−1)det(B−λ I)det(S) = det

(S−1S

)det(B−λ I)

= det(I)det(B−λ I) = det(B−λ I) (10.1)