Necessary and sufficient conditions for nonlinear matrix identities to always hold

Abstract

Matrix equations involving a family of unknown matrices $$X_1, \ldots , X_k$$ can be written in a general form $$f(X_1, \ldots , X_k) = 0$$ . One of the fundamental problems in matrix equation theory is to establish identifying conditions for the matrix equation to always hold for all matrices $$X_1, \ldots , X_k$$ . In this article, we select some nonlinear matrix equations with two, three, and four unknown matrices as illustrative examples, and show how to establish a family of simple and well-organized necessary and sufficient conditions for these matrix equations to always hold with respect to all unknown matrices in the equations. The methods, results, and facts in this article can be used to efficiently establish various nonlinear identities with variables in different noncommutative algebraic structures.