Rational Minimax Approximations for Matrix-Valued Functions: Existence, Optimality and Algorithms

In this paper, we study rational minimax approximation for continuous complex matrix-valued functions in the Frobenius norm, where all approximant entries share a common denominator. This generalizes classical scalar rational approximation, with applications in system modeling, microwave design, and nonlinear eigenvalue problems. We first prove the existence of such matrix-valued approximants on point sets dense in themselves, extending Walsh's foundational scalar result. Next, we establish characterizations of the local and global minimax approximants by deriving primal/dual matrix-valued Kolmogorov criteria and a Ruttan-type sufficient condition for global optimality. For analytic functions on a continuum, we link continuum minimax approximation to approximation on its boundary, and finite boundary samples via the maximum norm principle. We show that Ruttan's sufficient optimality condition provides a certificate under which a minimax approximant obtained from the boundary or from a discrete set of boundary nodes also solves the original continuum problem. Finally, for discrete approximation, we connect these conditions to a dual problem and the related dual-based numerical method m-d-Lawson: when the original minimax problem admits a solution, strong duality is equivalent to Ruttan's sufficient optimality condition, and, the optimality equations underlying the m-d-Lawson iteration coincide with Kolmogorov's dual criteria. These results provide a theoretical basis for certifying and computing matrix-valued rational minimax approximants.