Exchange matrix and positive definiteness

[randommacuser]

Homework Statement

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.

Homework Equations

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.

 

[morphism]

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.

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.

 

[randommacuser]

Beautiful. Thanks for your help.