Eigenvalues for a matrix with equal and opposite diagonal entries?

[AlphaNumeric2]

Given a square matrix (arbitrary finite size) where two diagonal entries are 'a' and '-a', what can you derive about the eigenvalues of the matrix?

My supervisor mentioned she'd read something about it being provable that the matrix cannot be positive or negative definite. Two of the eigenvalues will certainly have opposite signs or at most be both zero. She says she read it on a book but has completely forgotten where or what the result's name was (if it has one).

Can anyone confirm this result? The square matrix is completely general except for two diagonal entries being the negative of one another. I'm not sure if they had to be successive diagonal entries. If it is true, what's the proof? If it's too long a proof or requires a bunch of lemma to build up to it from more well known results, can someone point me to a book or website which covers it?

Thanks :)

 

[matt grime]

Have you considered writing down 2x2 matrices with diagonal entries 1 and -1 and playing around with things? It is very illuminating, and should have been the first thing you did.

 

[MathematicalPhysicist]

well obviously the charectaristic polynomial will be: P(x)=(x^2-a^2)+bc=x^2+(bc-a^2) but it seems to be triviality thing to check is it not?

 

[mathwonk]

so what's the trace?