Correlation between Two Projected Matrices Under Isometry Constraints

Abstract

In this paper, we consider the problem of correlation between the projections of two square matrices. These matrices of dimensions m x m and n x n are projected on a subspace of lower-dimension k under isometry constraints. We maximize the correlation between these projections expressed as a trace function of the product of the projected matrices. First we connect this problem to notions such as the generalized numerical range, the field of values and the similarity matrix. We show that these concepts are particular cases of our problem for choices of m, n and k. The formulation used here applies to both real and complex matrices. We characterize the objective function, its fixed points, its optimal value for Hermitian and normal matrices and finally upper and lower bounds for the general case. An iterative algorithm based on the singular value decomposition is proposed to solve the optimization problem.

Keywords: correlation, trace maximization, generalized numerical range, isometry