In Section 5.1 we discussed how to decide whether a given number is an eigenvalue of a matrix, and if so, how to find all of the associated eigenvectors. In this section, we will give a method for computing all of the eigenvalues of a matrix. This does not reduce to solving a system of linear equations: indeed, it requires solving a nonlinear equation in one variable, namely, finding the roots of the characteristic polynomial.

Definition

Let be an matrix. The characteristic polynomial of is the function given by

We will see below that the characteristic polynomial is in fact a polynomial. Finding the characterestic polynomial means computing the determinant of the matrix whose entries contain the unknown

It is time that we justified the use of the term “polynomial.” First we need a vocabulary word.

Definition

The trace of a square matrix is the number obtained by summing the diagonal entries of

Theorem

Let be an matrix, and let be its characteristic polynomial. Then is a polynomial of degree Moreover, has the form

In other words, the coefficient of is and the constant term is (the other coefficients are just numbers without names).

Proof

First we notice that

so that the constant term is always

We will prove the rest of the theorem only for matrices; the reader is encouraged to complete the proof in general using cofactor expansions. We can write a matrix as then

Recipe: The characteristic polynomial of a matrix

When the previous theorem tells us all of the coefficients of the characteristic polynomial:

This is generally the fastest way to compute the characteristic polynomial of a matrix.

By the above theorem, the characteristic polynomial of an matrix is a polynomial of degree Since a polynomial of degree has at most roots, this gives another proof of the fact that an matrix has at most eigenvalues. See this important note in Section 5.1.

Eigenvalues of a triangular matrix

It is easy to compute the determinant of an upper- or lower-triangular matrix; this makes it easy to find its eigenvalues as well.

Corollary

If is an upper- or lower-triangular matrix, then the eigenvalues of are its diagonal entries.

Proof

Suppose for simplicity that is a upper-triangular matrix:

Its characteristic polynomial is

This is also an upper-triangular matrix, so the determinant is the product of the diagonal entries:

If is an matrix, then the characteristic polynomial has degree by the above theorem. When one can use the quadratic formula to find the roots of There exist algebraic formulas for the roots of cubic and quartic polynomials, but these are generally too cumbersome to apply by hand. Even worse, it is known that there is no algebraic formula for the roots of a general polynomial of degree at least

In practice, the roots of the characteristic polynomial are found numerically by computer. That said, there do exist methods for finding roots by hand. For instance, we have the following consequence of the rational root theorem (which we also call the rational root theorem):

Rational Root Theorem

Suppose that is an matrix whose characteristic polynomial has integer (whole-number) entries. Then all rational roots of its characteristic polynomial are integer divisors of

For example, if has integer entries, then its characteristic polynomial has integer coefficients. This gives us one way to find a root by hand, if has an eigenvalue that is a rational number. Once we have found one root, then we can reduce the degree by polynomial long division.

In the above example, we could have expanded cofactors along the second column to obtain

Since was the only nonzero entry in its column, this expression already has the term factored out: the rational root theorem was not needed. The determinant in the above expression is the characteristic polynomial of the matrix so we can compute it using the trace and determinant:

Let be an matrix. Here are some strategies for factoring its characteristic polynomial First, you must find one eigenvalue:

Do not multiply out the characteristic polynomial if it is already partially factored! This happens if you expand cofactors along the second column in this example.

If there is no constant term, you can factor out as in this example.

If the matrix is triangular, the roots are the diagonal entries.

Guess one eigenvalue using the rational root theorem: if is an integer, substitute all (positive and negative) divisors of into

Find an eigenvalue using the geometry of the matrix. For instance, a reflection has eigenvalues

After obtaining an eigenvalue use polynomial long division to compute This polynomial has lower degree. If then this is a quadratic polynomial, to which you can apply the quadratic formula to find the remaining roots.