Selected Nonlinear Matrix Equations

The aim of this book is to present results about nonlinear matrix equations, which can be useful to anyone with interest in matrix equations. This book will also give ideas to graduate students working on matrix equations or in numerical linear algebra. In this book four types of nonlinear matrix equations are considered: Nonsymmetric Algebraic Riccati Equation, for which it is shown, that when the coefficient matrices form an irreducible M- matrix, there exists a second positive solution, the form of the second positive solution and how to find it is also specified. Second type is Nonsymmertic Matrix Riccati Differential Equation - it is shown, that when the coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix the equation has a global solution and for a wide range of initial values the global solution converges to the stable equilibrium solution. In the second part of the book the plus and minus equations are considered. An iterative method approximating the maximal positive definite solution is analyzed and the underlying reason for the speed up of convergence is revealed.