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
Then every eigenvalue is contained in some Gerschgorin disc.
This theorem says to add up the absolute values of the entries of the ith row which are off the main
diagonal and form the disc centered at aii having this radius. The union of these discs contains
Proof: Suppose Ax = λx where x≠0. Then for A =
Now dividing by
, it follows
is contained in the kth
Gerschgorin disc. ■
Example 14.3.2 Here is a matrix. Estimate its eigenvalues.
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.
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
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