Introduction to the Mathematical and Statistical Foundations of Econometrics
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. 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”).
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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.
