# Eigenvalue - geometric multiplicity proof

1. Sep 30, 2009

### flashdrive

1. The problem statement, all variables and given/known data

Given matrix A:

a 1 1 ... 1
1 a 1 ... 1
1 1 a ... 1
.. . .. ... 1
1 1 1 ... a

Show there is an eigenvalue of A whose geometric multiplicity is n-1. Express its value in terms of a.

2. Relevant equations

general eigenvalue/vector equations

3. The attempt at a solution

My problem is I'm not sure how to start it off.
I can state A is square, symmetric and hermitian so I know it has to do with one or more of those. I tried going through using a determinant but it didn't seem to work nicely, I have a feeling that it might have to do with a property of symmetric matrices but am not sure how to go about the proof (or which property)

2. Sep 30, 2009

### HallsofIvy

Staff Emeritus
Well, first, what is the eigenvalue in question? To do that, of course, you will need to find the characteristic polynomial. I recommend starting with "1 by 1", "2 by 2", and "3 by 3" matrices to see if you can find a pattern. Do the same thing to find the eigenvectors corresponding to that eigenvalue.