André C. M. Ran

A fixed point theorem in partially ordered sets and some applications to matrix equations

An analogue of Banachs fixed point theorem in partially ordered sets is proved in this paper, and several applications to linear and nonlinear matrix equations are discussed.

Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + A*X-1A = Q

Abstract We consider the problem of when the matrix equation X + A ∗ X -1 A = Q has a positive definite solution. Here Q is positive definite. We study both the real and the complex case. This equation plays a crucial role in solving a special case of the discrete-time Riccati equation. We present both necessary and sufficient conditions for its solvability. This result is obtained by using an analytic factorization approach. Moreover, we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation, respectively. Finally, we discuss the number of solutions.

Existence and Comparison Theorems for Algebraic Riccati Equations for Continuous- and Discrete-Time Systems

Abstract We discuss two comparison theorems for algebraic Riccati equations of the form XBR -1 B ∗ X−X(A−BR -1 C) -(A−BR -1 C) ∗ X−(Q−C ∗ R -1 C) = 0. Simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a Riccati equation from the existence of a hermitian solution of a second Riccati equation. Further, similar results are given for discrete algebraic Riccati equations.

On an Iteration Method for Solving a Class of Nonlinear Matrix Equations

This paper treats a set of equations of the form

On the nonlinear matrix equation X+A∗F(X)A=Q: solutions and perturbation theory

X+A^{\star}{\cal F}(X)A =Q

Stability of Invariant Lagrangian Subspaces II

, where

Stability of invariant maximal semidefinite subspaces. I

{\cal F}

Local inverse spectral problems for rational matrix functions

maps positive definite matrices either into positive definite matrices or into negative definite matrices, and satisfies some monotonicity property. Here A is arbitrary and Q is a positive definite matrix. It is shown that under some conditions an iteration method converges to a positive definite solution. An estimate for the rate of convergence is given under additional conditions, and some numerical results are given. Special cases are considered, which cover also particular cases of the discrete algebraic Riccati equation.

Hermitian solutions of the discrete algebraic Riccati equation

Abstract In this paper, the nonlinear matrix equation X+A ∗ F (X)A=Q is discussed. Sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived. Also conditions are given under which the solution depends continuously on the matrices A and Q .

Polar decompositions in finite dimensional indefinite scalar product spaces: General theory

Recently, the problems of stability of invariant subspaces of matrices and operators, i.e. the behaviour of invariant subspaces under small perturbations of the matrix or the operator, attracted much attention (see [BGK, GR, GLR1, AFS]). The main motivation to consider these problems comes from factorizations of matrix and operator functions, where invariant subspaces appear as the main tool in describing the factorizations (for this approach to factorization see, e.g., [BGK, GLR2]).