Extremal ranks of a quadratic matrix expression with applications

摘要

In this article, we give some closed-form formulae for the maximal and minimal ranks of the quadratic matrix expression q(X_1,X_2)= A-(A_1-B_1X_1C_1)D(A_2-B_2X_2C_2) with respect to the two variable matrices X_1 and X_2. As an application, we give the maximal and minimal ranks of the Schur complement D - CA-r B with respect to the reflexive generalized inverse A-r of A. In addition, we derive necessary and sufficient conditions for the solutions of two linear matrix equations to be orthogonal and proportional, respectively.