A continuous Jacobi-like approach to the simultaneous reduction of real matrices

Abstract

The problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered. Based on the Jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced, the projected-gradient method is used in this paper. It is shown that the projected gradient of the objective function can be formulated explicitly. This gives rise to a system of ordinary differential equations that can be readily solved by numerical software. The advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary, and that if a desired form is not attainable, then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form. The general procedure for deriving these differential equations is discussed. The framework can be generalized to the complex-valued case. Some applications are given.