# Homework Help: Orthogonal Projection onto Hilbert Subspace

1. Sep 3, 2010

### Kreizhn

1. The problem statement, all variables and given/known data
I have a fixed unitary matrix, say $X_d \in\mathfrak U(N)$ and a skew Hermitian matrix $H \in \mathfrak u(N)$. Consider the trace-inner product
[tex] \langle A,B \rangle = \text{Tr}[A^\dagger B ] [/itex]
where the dagger is the Hermitian transpose. I'm trying to find the orthogonal projection of $X_d$ onto the space
[tex] S = \left\{ X \ : \ X = \exp[tH], t \in \mathbb R \right\} [/itex]

3. The attempt at a solution
It seems to me that this problem shouldn't be terribly difficult. The notion of orthogonal projections in Hilbert spaces is well studied. However, I need a concrete value (or even better, a general projection operator) that takes $X_d$ to the point "closest" in S.

Obviously, the distance set-point distance is
[tex] d(X_d, S) = \inf_{X\in S,} \langle X-X_d,X-X_d \rangle [/itex]
but I don't see how this should give me the projection itself. In particular, this is a one dimensional subspace so does that make it easier to find somehow? Hopefully somebody out there can give me a push in the correct direction.