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Orthogonal Projection onto Hilbert Subspace

  1. Sep 3, 2010 #1
    1. The problem statement, all variables and given/known data
    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]

    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 [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.
  2. jcsd
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