Extremal ranks of a quadratic matrix expression with applications
Yongge Tian · Linear and Multilinear Algebra · 2011
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 1 X 1 C 1)D(A 2 − B 2 X 2 C 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 with respect to the reflexive generalized inverse of A. In addition, we derive necessary and sufficient conditions for the solutions of two linear matrix equations to be orthogonal and proportional, respectively.