Extremal Ranks of Some Nonlinear Matrix Expressions with Applications

Abstract

In this paper, we establish a group of closed-form formulas for the maximal and minimal ranks of a nonlinear matrix expression with respect to two variant matrices by using a linearization method and some known formulas for extremal ranks of linear matrix expressions. In addition, by using some pure algebraic operations of matrices and their generalized inverses, we derive the maximal and minimal ranks of the above nonlinear matrix expression, where the two variant matrices are any solutions of two consistent matrix equations. As an application, we derive some sufficient and necessary conditions for the existence of the solution of a nonlinear matrix function.