First Problem: Assume, as the problem indicates, that is Hermitian and invertible. Then we know that

for all

Let Then, because the inverse exists, we may define and such that both Now play around with

You'd like it to equal

Can you get that to happen?

Second Problem. Let me rephrase your question using more standard notation.

is a linear space with inner product

Let be the subspace consisting of the following:

Let be defined by

Prove that is skew-symmetric.

Is this a correct re-statement of the problem? If so, I have an extremely strong feeling that integration by parts is going to be the key to solving this problem. The subspace you're in indicates that the typical boundary term of

will be zero, which means you'll just pick up that minus sign when you try to slap the integral on the other term. Try that and see if it doesn't get you where you need to go.