There are ways to estimate the eigenvalues for matrices. The most famous is known as
Gerschgorin’s theorem. This theorem gives a rough idea where the eigenvalues are just from
looking at the matrix.

Theorem 6.8.1Let A be an n × n matrix. Consider the n Gerschgorin discs definedas

({ )}
D ≡ λ ∈ ℂ : |λ− a | ≤ ∑ |a | .
i ( ii j⁄=i ij )

Then every eigenvalue is contained in some Gerschgorin disc.

This theorem says to add up the absolute values of the entries of the i^{th} row which are off the
main diagonal and form the disc centered at a_{ii} having this radius. The union of these discs
contains σ

(A)

.

Proof: Suppose Ax = λx where x≠0. Then for A =

(aij)

, let

|xk|

≥

|xj|

for all x_{j}. Thus

|xk|

≠0.

∑
akjxj = (λ − akk)xk.
j⁄=k

Then

|| ||
|x |∑ |a | ≥ ∑ |a ||x | ≥ ||∑ a x || = |λ − a ||x |.
k j⁄=k kj j⁄=k kj j ||j⁄=k kj j|| ii k

Now dividing by

|xk|

, it follows λ is contained in the k^{th} Gerschgorin disc. ■

Example 6.8.2Here is a matrix. Estimate its eigenvalues.

( )
2 1 1
|( 3 5 0 |)
0 1 9

According to Gerschgorin’s theorem the eigenvalues are contained in the disks

It is important to observe that these disks are in the complex plane. In general this is the case. If
you want to find eigenvalues they will be complex numbers.

PICT

So what are the values of the eigenvalues? In this case they are real. You can compute them by
graphing the characteristic polynomial, λ^{3}− 16λ^{2} + 70λ− 66 and then zooming in on the zeros. If
you do this you find the solution is

{λ = 1.2953}

,

{λ = 5.5905}

,

{λ = 9.1142}

. Of course these
are only approximations and so this information is useless for finding eigenvectors. However, in
many applications, it is the size of the eigenvalues which is important and so these numerical
values would be helpful for such applications. In this case, you might think there is no real reason
for Gerschgorin’s theorem. Why not just compute the characteristic equation and graph and
zoom? This is fine up to a point, but what if the matrix was huge? Then it might be hard
to find the characteristic polynomial. Remember the difficulties in expanding a big
matrix along a row or column. Also, what if the eigenvalues were complex? You don’t see
these by following this procedure. However, Gerschgorin’s theorem will at least estimate
them.