Orthogonal Projection onto Hilbert Subspace

[Kreizhn]

Homework Statement

I have a fixed unitary matrix, say [itex] X_d \in\mathfrak U(N)[/itex] and a skew Hermitian matrix [itex] H \in \mathfrak u(N) [/itex]. 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 [itex] X_d [/itex] onto the space
[tex] S = \left\{ X \ : \ X = \exp[tH], t \in \mathbb R \right\} [/itex]

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 [itex] X_d [/itex] 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.

 

[mighty2000]

Thank you for posting your question. I am happy to help you with this problem.

Firstly, let's define the orthogonal projection operator onto a subspace. In general, the projection of a vector v onto a subspace S is given by P_S(v) = u, where u is the closest vector to v in S. In other words, u is the vector in S that minimizes the distance ||v-u||.

In your case, the subspace S is defined as the set of matrices X = exp(tH) where t is a real number. Now, let's consider the projection of X_d onto this subspace. We want to find the matrix X in S that minimizes the distance ||X-X_d||.

Using the trace-inner product, we can rewrite this distance as ||X-X_d||^2 = <X-X_d, X-X_d> = Tr[(X-X_d)^\dagger (X-X_d)]. Expanding this out, we get Tr[X^\dagger X + X_d^\dagger X_d - X^\dagger X_d - X_d^\dagger X].

Now, since we want to minimize this distance, we can take the derivative with respect to X and set it equal to 0. This will give us the matrix X that minimizes the distance. Taking the derivative and setting it equal to 0, we get X = X_d exp(-tH).

So, the projection of X_d onto the subspace S is given by P_S(X_d) = X_d exp(-tH), where t is a real number. This is the closest matrix to X_d in S, and is the solution to your problem.

I hope this helps you with your problem. Feel free to ask any further questions if needed.
Scientist