scholarly journals Symmetric Self-adjunctions and Matrices

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1051-1082 ◽  
Author(s):  
Kosta Došen ◽  
Zoran Petrić
Keyword(s):  
Kronecker Product ◽  
The Self ◽  
Adjoint Functor ◽  
Centralizer Algebras ◽  
Product Of Matrices

It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices.

scholarly journals Least Squares Pure Imaginary Solution and Real Solution of the Quaternion Matrix EquationAXB+CXD=Ewith the Least Norm

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shi-Fang Yuan
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Kronecker Product ◽  
Generalized Inverse ◽  
Complex Representation ◽  
Real Solution ◽  
Quaternion Matrix ◽  
Quaternion Matrices ◽  
Quaternion Matrix Equation ◽  
Product Of Matrices

Using the Kronecker product of matrices, the Moore-Penrose generalized inverse, and the complex representation of quaternion matrices, we derive the expressions of least squares solution with the least norm, least squares pure imaginary solution with the least norm, and least squares real solution with the least norm of the quaternion matrix equationAXB+CXD=E, respectively.

scholarly journals The least squares η-Hermitian problems of quaternion matrix equation AHXA + BHYB=C

Filomat ◽  
2014 ◽  
Vol 28 (6) ◽  
pp. 1153-1165 ◽  
Author(s):  
Shi-Fang Yuan ◽  
Qing-Wen Wang ◽  
Zhi-Ping Xiong
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Kronecker Product ◽  
Generalized Inverse ◽  
Hermitian Matrix ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Quaternion Matrices ◽  
Quaternion Matrix Equation ◽  
Product Of Matrices

For any A=A1+A2j?Qnxn and ?? {i,j,k} denote A?H = -?AH?. If A?H = A,A is called an ?-Hermitian matrix. If A?H =-A,A is called an ?-anti-Hermitian matrix. Denote ?-Hermitian matrices and ?-anti-Hermitian matrices by ?HQnxn and ?AQnxn, respectively. In this paper, we consider the least squares ?-Hermitian problems of quaternion matrix equation AHXA+ BHYB = C by using the complex representation of quaternion matrices, the Moore-Penrose generalized inverse and the Kronecker product of matrices. We derive the expressions of the least squares solution with the least norm of quaternion matrix equation AHXA + BHYB = C over [X,Y] ? ?HQnxn x ?HQkxk, [X,Y] ? ?AQnxn x ?AQkxk, and [X,Y] ? ?HQnxn x ?AQkxk, respectively.

Least Squares Skew Bisymmetric Solution for a Kind of Quaternion Matrix Equation

2011 ◽  
Vol 50-51 ◽  
pp. 190-194 ◽  
Author(s):  
Shi Fang Yuan ◽  
Han Dong Cao
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Kronecker Product ◽  
Special Structure ◽  
Complex Representation ◽  
Quaternion Matrix ◽  
Quaternion Matrices ◽  
Quaternion Matrix Equation ◽  
Product Of Matrices

In this paper, by using the Kronecker product of matrices and the complex representation of quaternion matrices, we discuss the special structure of quaternion skew bisymmetric matrices, and derive the expression of the least squares skew bisymmetric solution of the quaternion matrix equation AXB =C with the least norm.

scholarly journals h-Stability of Linear Matrix Differential Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Peiguang Wang ◽  
Xiaowei Liu
Keyword(s):  
Stability Problem ◽  
Kronecker Product ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix System ◽  
Differential Systems ◽  
Perturbed System ◽  
Matrix Differential ◽  
The Stability ◽  
Product Of Matrices

This paper investigates the stability problem of linear matrix differential systems and gives some sufficient conditions ofh-stability for linear matrix system and its associated perturbed system by using the Kronecker product of matrices. An example is also worked out to illustrate our results.

scholarly journals Multidimensional Arrays, Indices and Kronecker Products

Econometrics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 18
Author(s):  
D. Stephen G. Pollock
Keyword(s):  
Matrix Algebra ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Multidimensional Arrays ◽  
Conventional Notation ◽  
Index Notation ◽  
Product Of Matrices

Much of the algebra that is associated with the Kronecker product of matrices has been rendered in the conventional notation of matrix algebra, which conceals the essential structures of the objects of the analysis. This makes it difficult to establish even the most salient of the results. The problems can be greatly alleviated by adopting an orderly index notation that reveals these structures. This claim is demonstrated by considering a problem that several authors have already addressed without producing a widely accepted solution.

scholarly journals On the Ψ-asymptotic equivalence of the Ψ-bounded solutions of two Lyapunov matrix differential equations

2020 ◽  
Vol 34 ◽  
pp. 03006
Author(s):  
Aurel Diamandescu
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Kronecker Product ◽  
Existence Results ◽  
Bounded Solutions ◽  
Asymptotic Equivalence ◽  
Matrix Differential Equations ◽  
Lyapunov Matrix ◽  
Matrix Differential ◽  
Product Of Matrices

Using Schauder Tychonoff fixed point theorem and the technique of Kronecker product of matrices, we prove existence results for Ψ-asymptotic equivalence of the Ψ-bounded solutions of two Lyapunov matrix differential equations.

scholarly journals Kronecker Product of matrices and their applications to self-adjoint two-point boundary value problems associated with first order matrix differential systems

2021 ◽  
Vol 10 (10) ◽  
pp. 25399-25407
Author(s):  
Sriram Bhagavatula ◽  
Dileep Durani Musa ◽  
Murty Kanuri
Keyword(s):  
Boundary Value Problems ◽  
Kronecker Product ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Least Square ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Differential Systems ◽  
First Order ◽  
Product Of Matrices

In this paper, we shall be concerned with Kronecker product or Tensor product of matrices and develop their properties in a systematic way. The properties of the Kronecker product of matrices is used as a tool to establish existence and uniqueness of solutions to two-point boundary value problems associated with system of first order differential systems. A new approach is described to solve the Kronecker product linear systems and establish best least square solutions to the problem. Several interesting examples are given to highlight the importance of Kronecker product of matrices. We present adjoint boundary value problems and deduce a set of necessary and sufficient conditions for the Kronecker product boundary value problem to be self-adjoint.

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