This research will concern itself with several of the core areas of linear algebra: matrix equations, inertia, and stability, which have relevance both to computational linear algebra and to control theory. First the study of conditions for nonsingularity of solutions of Sylvester, Lyapunov, and Riccati matrix equations will be studied. Second the principal investigator plans to study orthogonal solutions of matrix equations. Third the principal investigator plans to study the construction of symmetrizers with specific properties of use in applications. Fourth the principal investigator will study several areas of inertia theory, namely singular inertia theory, inertia theory for Riccati equations, and generalized inertia. Finally the principal investigator will study when positive definite matrices have positive definite and semidefinite images under certain Lyapunov maps.
