OPTIMAL RANKS OF NONLINEAR MATRIX EXPRESSIONS WITH APPLICATIONS

Abstract

This paper presents explicit formulas for the optimal ranks of nonlinear matrixexpressions with respect to four variable matrices. As applications, we derive the optimalranks of the generalized Schur complement and present the optimal ranks of specific non-linear matrix expressions involving four variable matrices, which represent solutions to foursolvable matrix equations. Consequently, we provide equivalent conditions for the existenceof solutions to nonlinear matrix equations.