## Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are scalars and vectors associated with any square

matrix and identify key properties of the matrix. They have widespread use in

economics, as illustrated by the following examples. Consider a dynamic system

of equations, yt+1 = Ayt, where A is an arbitrary matrix with real entries. Over time, the evolution of the system is governed by powers of A: yt+ j = A j yt. This formulation leads naturally to the study of powers of a matrix A with real entries

and this task is greatly simplified by using eigenvectors and eigenvalues of A.

As a second example, Markov processes evolve according to a transition matrix T

according to xt+1 = xtT. In that case the matrix consists of non-negative entries and each row sums to 1. A steady state of this system is a vector x¯ with x¯ = x¯T. It turns out that this steady state is associated with a specific eigenvalue of the

matrix T. In least squares estimation, the estimated errors, e, are related to the

dependent variable, y, according to a formula e = M y. Given the statistical distribution of y, the distribution of e is determined by M, and in the study of M,

its eigenvalues and eigenvectors play a useful role. Finally, in certain optimiza-

tion problems, location of an optimum is closely related to a particular symmetric

matrix having the property that xAx > 0 for all x. The study of each of these problems is facilitated through the use of eigenvalues and eigenvectors.