Extrema of matrix functionals

The problem of determining conditional extrema of functionals with matrix arguments is considered. We derive the necessary and sufficient mathematical conditions for the existence of extrema of functionals satisfying constraints of the form of matrix equalities on the arguments. The construction of extrema is based on functions and matrices of indeterminate Lagrange multipliers. As applications we consider an example of determining the optimal strength coefficient matrix in a dynamical system with an adaptive Carleman filter and an example, from the theory of statistical decisions, of minimizing the volume of the dispersion error ellipsoid. Our approach has wide applications not only in optimization problems from automatic control theory but also in mathematical statistics and the theory of material strength and plasticity.Translated from Dinamicheskie Sistemy, No. 5, pp. 103–106, 1986.