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.