Hi, I have an applied statistics problem regarding invariant subspaces, which I am unable to solve. Perhaps someone can help me? I am trying to minimise a function f = ln |U'*C*U| + trace( (U'*C*U)^(-1) (U'*S*U) ) with respect to the nxp matrix U (p<n), subject to the constraint that U'*U=I. C and S are both full-rank nxn matrices. After applying some horrible matrix calculus using Lagrangian multipliers, I am able to determine that a necessary condition on the minimum is U*U'*C*U = C*U + S*U - C*U (U'*C*U)^(-1) U'*S*U. Evidently only the subspace spanned by U is unique, since U*R also satisfies the equations for any orthogonal R. This observation also makes it possible to assume either C or S diagonal without any loss of generality. I can also simultaneously diagonalise them (and even make one of them identity), but only at the cost of complicating the constraint. Even with these insights, a method of solution has completely eluded me. I can't even seem to solve it for the apparently simpler case where U is a nx1 vector. Any suggestions would be extremely, extremely welcome. Thanks, Fred
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