scholarly journals Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Li Wang
Keyword(s):  
Riccati Equation ◽  
Positive Semidefinite ◽  
Algebraic Riccati Equation ◽  
Positive Semidefinite Matrix ◽  
Matrix Inequalities ◽  
Numerical Examples ◽  
Upper Solution ◽  
Solution Bound ◽  
Semidefinite Matrix ◽  
Two Parameter

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

scholarly journals Matrix inequalities from a two variables functional

2016 ◽  
Vol 27 (09) ◽  
pp. 1650071 ◽  
Author(s):  
Jean-Christophe Bourin ◽  
Eun-Young Lee
Keyword(s):  
Triangle Inequality ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Matrix Inequalities ◽  
Trace Inequality ◽  
Positive Linear Map ◽  
Normal Operators ◽  
Matrix Formula ◽  
Majorization Theorem ◽  
Semidefinite Matrix

We introduce a two variables norm functional and establish its joint log-convexity. This entails and improves many remarkable matrix inequalities, most of them related to the log-majorization theorem of Araki. In particular: if[Formula: see text] is a positive semidefinite matrix and[Formula: see text] is a normal matrix,[Formula: see text] and[Formula: see text] is a subunital positive linear map, then[Formula: see text] is weakly log-majorized by[Formula: see text]. This far extension of Araki’s theorem (when [Formula: see text] is the identity and [Formula: see text] is positive) complements some recent results of Hiai and contains several special interesting cases, such as a triangle inequality for normal operators and some extensions of the Golden–Thompson trace inequality. Some applications to Schur products are also obtained.

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

scholarly journals A New Algorithm for Positive Semidefinite Matrix Completion

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Fangfang Xu ◽  
Peng Pan
Keyword(s):  
Matrix Completion ◽  
Positive Semidefinite ◽  
Alternating Direction Method ◽  
Low Rank ◽  
System Control ◽  
Positive Semidefinite Matrix ◽  
Linear Least Squares ◽  
Alternating Direction ◽  
Semidefinite Matrix Completion ◽  
Semidefinite Matrix

Positive semidefinite matrix completion (PSDMC) aims to recover positive semidefinite and low-rank matrices from a subset of entries of a matrix. It is widely applicable in many fields, such as statistic analysis and system control. This task can be conducted by solving the nuclear norm regularized linear least squares model with positive semidefinite constraints. We apply the widely used alternating direction method of multipliers to solve the model and get a novel algorithm. The applicability and efficiency of the new algorithm are demonstrated in numerical experiments. Recovery results show that our algorithm is helpful.

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