Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + A*X-1A = Q
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In this article, the authors considered the problem of finding a positive definite solution of the matrix equation X + A ∗ X -1 A = Q in a special case of the discrete-time Riccati equation.About:
This article is published in Linear Algebra and its Applications.The article was published on 1993-06-01 and is currently open access. It has received 182 citations till now. The article focuses on the topics: Riccati equation & Algebraic Riccati equation.read more
Citations
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Iterative solution of two matrix equations
TL;DR: Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation andumerical results are reported to illustrate the convergence behaviour of various algorithms.
Journal ArticleDOI
On an Iteration Method for Solving a Class of Nonlinear Matrix Equations
TL;DR: It is shown that under some conditions an iteration method converges to a positive definite solution of a set of equations of the form X+A^{\star}{\cal F}(X)A =Q, where A is arbitrary and Q is apositive definite matrix.
Journal ArticleDOI
On the positive definite solutions of the matrix equations Xs±ATX−tA=In☆
Xin-Guo Liu,Hua Gao +1 more
TL;DR: In this paper, the existence of symmetric positive definite solutions of the two matrix equations X s + A T X − t t A = I n and X s − A T x − t A= I n is proved based on the fixed-point theory.
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Hermitian solutions of the equation X = Q + NX−1N∗
Augusto Ferrante,Bernard C. Levy +1 more
TL;DR: In this article, the authors consider the matrix equation X = Q + NX −1 N∗, where the Hermitian solutions are parametrized in terms of the generalized Lagrangian eigenspaces of a certain matrix pencil.
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Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations
Wen-Wei Lin,Shu-Fang Xu +1 more
TL;DR: A unified convergence theory for the structure-preserving doubling algorithms for a class of Riccati-type matrix equations is established, using only elementary matrix theory.
References
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Journal ArticleDOI
An iterative technique for the computation of the steady state gains for the discrete optimal regulator
TL;DR: In this correspondence an algorithm is presented for computing the steady-state optimal feedback law of the discrete-time invariant linear regulator that converges quadratically in a neighborhood of the steady state.
Journal ArticleDOI
On the existence of a positive definite solution of the matrix equation X+ATX−1A= I
TL;DR: In this article, the question under which conditions on the real (square) matrix A the matrix equation X+ATX−1A=I has a real symmetric positive definite solution X is raised.
Journal ArticleDOI
Positive solutions to X = A−BX-1 B∗
TL;DR: In this article, the authors studied the positive (semidefinite) solutions to the matrix equation X = A− BX -1 B ∗ under the assumption that A⩾0.
Book ChapterDOI
Unitary Rational Matrix Functions
Daniel Alpay,Israel Gohberg +1 more
TL;DR: In this article, the theory of realization and minimal factorization of rational matrix valued functions unitary on the unit circle or on the imaginary line is presented in the framework of a indefinite scalar product.