An algorithm for maximizing a quotient of two Hermitian form determinants with different exponents

We investigate the maximization of a quotient of two determinants with different exponents under a Frobenius norm constraint, where each determinant is taken from a matrix-valued Hermitian form. The optimum matrix that constitutes the Hermitian forms is shown to be a scaled partial isometry. For the special case of vector-valued Hermitian forms, the optimality condition turns out to be an implicit eigenproblem and we derive an iterative algorithm where in each step the principal eigenvector of a matrix has to be chosen. In addition, we prove monotonic convergence of the iterative algorithm, which means that the utility increases in every step.