Linear algebra question involving symmetric matrices, a subspace of orthonormal vectors and their products and inverses.

01 January 2012 1 1K Report

I have attached a PDF which more elegantly describes this (hopefully) simple problem. In the course of developing a new theory as part of my graduate research I have encountered an expression U* (U^{T} * S * U)^{-1} *U^{T} where U contains random orthonormal vectors and S is positive-definite. Through brute force I have demonstrated that the product is invariant to the particular vectors one chooses to populate U, but I cannot prove why. Does anyone have any ideas?

Mike Specian

The answer is best viewed from the perspective of a projection matrix. Solution is in the attached PDF.

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