# Gershgorin Circle Theorem, mathematical derivation of eigenvalue estimates

1. Mar 28, 2012

### onako

I intend to use the Gershgorin Circle Theorem for estimating the eigenvalues of a real symmetric (n x n) matrix. Unfortunately, I'm a bit confused with the examples one might find on the internet; What would be the mathematical
formula for deriving estimates on eigenvalues?

I understand that certain disks are formed, each centered at the diagonal entry, with the radius equal to the summation of absolute values of the associated off-diagonal row entries. Which steps to take from this point to get the estimates on the eigenvalues?

2. Sep 10, 2017

### ExplosivePete

I hope you have found your solution 5 years later, but since others may have the same question and because I find the theorem very interesting and possibly useful, I'll post.

The Gershgorin disk theorem gives us that the spectrum of the matrix is a subset of the union of the gershgorin disks, i.e. the eigenvalues of the matrix are elements of the union of the disks as you rightly defined them. So the theorem gives an approximate value of the eigenvalues in the complex plane. Though in physics, we are often only dealing with real eigenvalues, so the disks are really intervals on the real number line. So your estimate would be any value in the interval.

I Hope this helps.

3. Sep 10, 2017

### Staff: Mentor

This thread has run its course. Normally older threads are closed, this one was open. So after a reasonable answer it seems to be time to shut the door.