# 12/2/2008

## Eigenvalues and eigenvectors

Consider an n x n matrix A. A nonzero vector v in R^n is called an eigenvector of A if Av is a scalar multiple of v. That is, Av = (lamda)v, for some scalar lamda. Note that the scalar lamda may be zero. The scalar lamda is called the eigenvalue associated with the eigenvector v.

## Important Question

How can we find eigenvalues and eigenvectors of an n x n matrix A?

Note lamda is an eigenvalue of A if and only if there is a nonzero vector v such that Av = (lamda) v.

This equation can be rewritten as Av - (lamda) v = 0 or Av - (lamda) I v = 0.

Notice that this is equivalent to saying that Ker(A - lamda I) is not equal to zero.

This statement is equivalent to saying that the matrix A - lamda I fails to be invertible or the determinant of A - lamda I is equal to zero.

In short, finding an eignevalue is equivalent to solving the characteristic equation det(A - lamda I) = 0.

## Discrete Dynamical Systems

Consider the dynamical system x(t+1) = A x(t) with the initial condition x(0) = x_0. Then x(t) = A^t x_0.

If we can find a basis of R^n that consists of the eigenvectors of A, then the state of the system at time t, x(t), can be written as a linear combination of the eigen values, eigenvectors, and coordinates of the initial vector x_0 with respect to the basis formed by the eigenvectors. (See Fact 7.1.3)

## Another characterization of an invertible matrix

An n x n matrix is invertible if (and only if) 0 fails to be an eigenvalue of A.

## Group/Class Exercises

Section 7.2 - 1, 3, 8, 11, 15, 17, 22, 28, 39

Section 7.1 - 1, 5, 8, 10, 24, 27, 31, 38, 51