## Eigenvectors

By Definition I.21 it follows that if M is an eigenvalue of an n x n matrix A, then A — MIn is a singular matrix (possibly complex valued!). Suppose first that M is real valued. Because the rows of A — MIn are linear dependent there exists a vector x є Kn such that (A — MIn)x — 0 (є Rn); hence, Ax — Mx. Such a

vector x is called an eigenvector of A corresponding to the eigenvalue M. Thus, in the real eigenvalue case:

Definition I.22: An eigenvector13 of an n x n matrix A corresponding to an eigenvalue M is a vector x such that Ax — Mx.

However, this definition also applies to the complex eigenvalue case, but then the eigenvector x has complex-valued components: x є Cn. To show the latter, consider the case that M is complex valued: M — a + i ■ в, а, в є К, в — 0...

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