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

Подпись: Aa1,1 a1,2

a2,1 a2,2

Подпись: a1,1 — X a1,2 a2, 1 a2, 2 — X
Подпись: det( A — XI2) = det

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:

Подпись: X1 =a1,1 + a2,2 + /(a17—a2~2f2+~4a12a22i

image886

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:

Подпись: MПодпись: M2a1,1 + 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!

Подпись: (a1,1 — a2,2)2 + 4a2,2
Подпись: a1 , 1 + a2, 2 + M1 =

Note that if the matrix A involved is symmetric (i. e., a1,2 — a2j1), then

Подпись: M2 =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.

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