Optimizing the Coupling Between Two Isometric Projections of Matrices

In this paper, we analyze the coupling between the isometric projections of two square matrices. These two matrices of dimensions $m\times m$ and $n\times n$ are restricted to a lower $k$-dimensional subspace under isometry constraints. We maximize the coupling between these isometric projections expressed as the trace 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 special choices of $m$, $n$, and $k$. The formulation used here applies to both real and complex matrices. We characterize the objective function, its critical points, and its optimal value for Hermitian and normal matrices, and, finally, give upper and lower bounds for the general case. An iterative algorithm based on the singular value decomposition is proposed to solve the optimization problem.