Correlation between orthogonally projected matrices
Catherine Fraikin, Yurii E. Nesterov, Paul Van Dooren · 2006
We consider the problem of finding the optimal correlation between two projected matrices U ∗ AU and V ∗ BV. The square matrices A and B may be of different dimensions, but the isometries U and V have a common column dimension k. The correlation is measured by the real function c(U,V) = ℜtr(U ∗ AUV ∗ B ∗ V), which we maximize of the isometries U ∗ U = V ∗ V = Ik. This problem can be viewed as an extension of the generalized numerical range of two matrices, which are now allowed to be of different dimension. We discuss several properties of this optimization problem, characterize its extremal points and propose an algorithm converging to such an extremal point.