two Hermitian problems
First problem: Assume T: V -> V is a Hermitian transformation
Prove T^-1 is Hermitian if T is invertible.
Here, I can prove that T^n is Hermitian if n > 0 but I'm stuck for n = -1.
Second Problem: C(0, 1) is linear Space. Inner product is given by:
(f, g) = integral( f * g, t, 0, 1).
Let V be the subspace of all f such that integral f, t, 0, 1) = 0.
Let T: V -> C(0, 1) and T(f(x)) = integral(f, t, 0, x). Prove T is skew - symmetric.
For this problem, I would apply the transformation, do the inner product, then I would have an integral in an integral. I don't know if I should F and G for their anti derivative or what.
First Problem: Assume, as the problem indicates, that is Hermitian and invertible. Then we know that
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.