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. Then

A M In A a In i ■ в In

12 Recall that the complex conjugate of x — a + i ■ b, a, b є К, is x — a — i ■ b. See Ap­pendix III.

13 Eigenvectors are also called characteristic vectors.

is complex valued with linear-dependent rows in the following sense. There existsavector x = a + i ■ b with a, b є К” andlength14 \x У = V aTa + bTb > 0 such that

(A — a In — і ■ в In )(a + і ■ b)

= [(A — a In )a + в b] + і ■ [(A — a In )b — в a] = 0(є К”).

Подпись: A — a In в In — в In A — a In image893 Подпись: eK2n. Подпись: (I.60)

Consequently, (A — a In )a + в b = 0 and (A — a In )b — в a = 0; thus,

Therefore, in order for the length of x to be positive, the matrix in (I.60) has to be singular; then Q can be chosen from the null space of this matrix.

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