The schur complement theorem and its applications

The quadratic matrix Riccati inequality arises in problems such as absolute stability, linear quadratic optimization, and optimal estimation. In this thesis, a known proof of Schur's Theorem concerning the positive definiteness of block matrices is presented, and using this theorem, the nonlinear matrix Riccati inequality is transformed into a linear matrix inequality by increasing the dimension. To solve the resulting inequality, a theorem related to the minimization of a maximum function – a more general problem – is proven. The obtained result is then applied not only to the Riccati inequality, but also to the Lyapunov inequality for a single matrix, the common quadratic Lyapunov function, and eigenvalue minimization problems. In solving the general minimization problem, norm bounds for the bounded functions involved are computed for various cases including the Lyapunov inequality, common Lyapunov function, Riccati inequality, and eigenvalue minimization problems. Keywords: Schur's theorem, Matrix Riccati inequality, Linear matrix inequality, Maximum function, Common Lyapunov function.