# Exchange matrix and positive definiteness

1. Aug 29, 2008

### randommacuser

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

Let E be the exchange matrix (ones on the anti-diagonal, zeroes elsewhere). Suppose A is symmetric and positive definite. Show that B = EAE is positive definite.

2. Relevant equations

3. The attempt at a solution

I've tried showing directly that for any conformable vector h, h'Bh > 0 whenever h'Ah > 0. This looks like a dead end. I suspect the easiest way to get the result is to show all the eigenvalues of B are positive, using the fact that all the eigenvalues of A are positive. However, I don't know how to show this.

2. Aug 29, 2008

### morphism

That's a good idea. Suppose k is an eigenvalue of B. Then EAEx=kx for some nonzero x. What happens if you multiply both sides by E?

Also note that for this to be a valid proof, you have to first show that B is symmetric, but this is trivial.

3. Aug 29, 2008

### randommacuser

Beautiful. Thanks for your help.