Linear Algebra and Analysis

14.3 The Estimation Of Eigenvalues

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 14.3.1 Let A be an n × n matrix. Consider the n Gerschgorin discs defined as

({ )} D \equiv λ \in ℂ : |λ - a | \leq \sum |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

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

, let

≥

for all x_j. Thus

≠0.

\sum akjxj = (λ- akk)xk. j⁄=k

Then

| | \sum \sum ||\sum || |xk| |akj| \geq |akj||xj| \geq || akjxj||= |λ- aii||xk|. j⁄=k j⁄=k |j⁄=k |

Now dividing by

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

Example 14.3.2 Here 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

D1 = {λ \in ℂ : |λ- 2| \leq 2},D2 = {λ \in ℂ : |λ- 5| \leq 3} ,

D3 = {λ \in ℂ : |λ- 9| \leq 1}

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.

So what are the values of the eigenvalues? In this case they are real. You can compute them by graphing the characteristic polynomial, λ³ − 16λ² + 70λ − 66 and then zooming in on the zeros. If you do this you find the solution is

. 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.