Introduction to the Mathematical and Statistical Foundations of Econometrics
Eigenvalues and Eigenvectors
I.15.1. Eigenvalues
Eigenvalues and eigenvectors play a key role in modern econometrics - in particular in cointegration analysis. These econometric applications are confined to eigenvalues and eigenvectors of symmetric matrices, that is, square matrices A for which A = AT. Therefore, I will mainly focus on the symmetric case.
Definition I.21: The eigenvalues11 ofann x n matrix A are the solutions for X of the equation det( A — X In) = 0.
It follows from Theorem I.29 that det(A) = J2 ±a1,i1 a2,i2... an, in, where the summation is over all permutations i1, i2,...,in of 1, 2,...,n. Therefore, if we replace A by A — XIn it is not hard to verify that det(A — XIn) is a polynomial of order n in X, det(A — XIn) = J^=o ckXk, where the coefficients ck are functions of the elements of A.
For example, in the 2 x 2 case
a1,1 a1,2
a2,1 a2,2
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we have
= (a1,1 — X)(a2,2 — X) — a1,2a2,1 = X2 — (a1,1 + a2,f)X + a1,1a2,2 — a1,2a2,b
which has two roots, that is, the solutions of X2 — (a1:1 + a2,2)X + a1,1a2,2 — a1,2a2,1 = 0:
a1,1 + a2,2 + /(a17—a2~2f2+~4a12a22i
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2
There are three cases to be distinguished. If (a1:1 — a2,2)2 + 4a1,2a2,1 > 0, then
11 Eigenvalues are also called characteristic roots. The name “eigen” comes from the German adjective eigen, which means “inherent,” or “characteristic.”
M and M2 are different and real valued. If (a1,1 — a2,2)2 + 4a1,2a2,1 — 0, then M = M2 and they are real valued. However, if (a1,1 — a2,2)2 + 4a1,2a2,1 < 0, then M and M2 are different but complex valued:
a1,1 + a2,2 + i ^/—(a21—'a2~2y2—~4ai~2a22[
2
a1,1 + a2,2 — i ^/—(«1,1 — a2,2)2 — 4a1,202,1
2
where i — V—T. In this case M1 and M2 are complex conjugate: M2 = ,M1.12 Thus, eigenvalues can be complex valued!
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Note that if the matrix A involved is symmetric (i. e., a1,2 — a2j1), then
a1,1 + a2,2 — У (a1,1 — a2,2)2 + 4a2,2
2
and thus in the symmetric 2 x 2 case the eigenvalues are always real valued. It will be shown in Section I.15.3 that this is true for all symmetric n x n matrices.
