A New Expression of the Hermitian Solutions to a System of Matrix Equations with Applications

In this paper, a new necessary and sufficient condition for the existence of a Hermitian solution as well as a new expression of the general Hermitian solution to the system of matrix equations A1X=C1 and A3XB3=C3 are derived. The max-min ranks and inertias of these Hermitian solutions with some interesting applications are shown. In particular, the max-min ranks and inertias of the Hermitian part of the general solution to this system are presented.

Download Full-text

A Note on Solutions of Linear Systems

ISRN Algebra ◽

10.1155/2013/142124 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Branko Malešević ◽

Ivana Jovović ◽

Milica Makragić ◽

Biljana Radičić

Keyword(s):

Linear System ◽

General Solution ◽

Linear Systems ◽

Sufficient Condition ◽

Matrix Equations ◽

Necessary And Sufficient Condition ◽

Different Types ◽

Free Parameters ◽

Necessary And Sufficient

We will consider Rohde's general form of {1}-inverse of a matrix A. The necessary and sufficient condition for consistency of a linear system Ax=c will be represented. We will also be concerned with the minimal number of free parameters in Penrose's formula x=A(1)c+(I-A(1)A)y for obtaining the general solution of the linear system. These results will be applied for finding the general solution of various homogenous and nonhomogenous linear systems as well as for different types of matrix equations.

Download Full-text

Ranks of a Constrained Hermitian Matrix Expression with Applications

Journal of Applied Mathematics ◽

10.1155/2013/514984 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Shao-Wen Yu

Keyword(s):

Hermitian Matrix ◽

Schur Complement ◽

Matrix Equations ◽

Quaternion Matrix ◽

Necessary And Sufficient Condition ◽

Common Solution ◽

System Of Matrix Equations ◽

Hermitian Solution ◽

Matrix Expression ◽

Necessary And Sufficient

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expressionC4−A4XA4∗whereXis a Hermitian solution to quaternion matrix equationsA1X=C1,XB1=C2, andA3XA3*=C3. As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equationsA1X=C1,XB1=C2,A3XA3*=C3, andA4XA4*=C4, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complementC4−A4A3~A4∗with respect to a Hermitian g-inverseA3~ofA3, which is a common solution to quaternion matrix equationsA1X=C1andXB1=C2, are also considered.

Download Full-text

A System of Matrix Equations over the Quaternion Algebra with Applications

Algebra Colloquium ◽

10.1142/s100538671700013x ◽

2017 ◽

Vol 24 (02) ◽

pp. 233-253 ◽

Cited By ~ 8

Author(s):

Xiangrong Nie ◽

Qingwen Wang ◽

Yang Zhang

Keyword(s):

General Solution ◽

Quaternion Algebra ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Matrix Equations ◽

Electronic Networks ◽

Consistency Conditions ◽

Numerical Example ◽

System Of Matrix Equations ◽

Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

Download Full-text

General Solution and Observability of Singular Differential Systems with Delay

Abstract and Applied Analysis ◽

10.1155/2013/512465 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Jiang Wei

Keyword(s):

General Solution ◽

Sufficient Condition ◽

General Description ◽

Necessary And Sufficient Condition ◽

Differential Systems ◽

Exact Observability ◽

Necessary And Sufficient

We study singular differential systems with delay. A general description for the solutions of singular differential systems with delay is given and a necessary and sufficient condition for exact observability of singular differential systems with delay is derived.

Download Full-text

On the Centro-symmetric Solution of a System of Matrix Equations over a Regular Ring with Identity

Algebra Colloquium ◽

10.1142/s100538670700051x ◽

2007 ◽

Vol 14 (04) ◽

pp. 555-570 ◽

Cited By ~ 7

Author(s):

Qingwen Wang ◽

Haixia Chang ◽

Chunyan Lin

Keyword(s):

General Solution ◽

Explicit Expression ◽

Symmetric Solution ◽

Regular Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Matrix Equations ◽

System Of Matrix Equations ◽

Necessary And Sufficient

In this paper, we find the centro-symmetric solution of a system of matrix equations over an arbitrary regular ring [Formula: see text] with identity. We first derive some necessary and sufficient conditions for the existence and an explicit expression of the general solution of the system of matrix equations A1X1 = C1, A2X1 = C2, A3X2 = C3, A4X2 = C4 and A5X1B5 + A6X2B6 = C5 over [Formula: see text]. By using the above results, we establish two criteria for the existence and the representation of the general centro-symmetric solution of the system of matrix equations AaX = Ca, AbX = Cb and AcXBc = Cc over the ring [Formula: see text].

Download Full-text

Solving Optimization Problems on Hermitian Matrix Functions with Applications

Journal of Applied Mathematics ◽

10.1155/2013/593549 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11

Author(s):

Xiang Zhang ◽

Shu-Wen Xiang

Keyword(s):

Optimization Problems ◽

Hermitian Matrix ◽

Sufficient Conditions ◽

Positive Definite ◽

Matrix Functions ◽

Matrix Equations ◽

Necessary And Sufficient Condition ◽

System Of Matrix Equations ◽

The Matrix ◽

Necessary And Sufficient

We consider the extremal inertias and ranks of the matrix expressionsf(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, whereA3=A3*, B3, C3, andD3are known matrices andYandXare the solutions to the matrix equationsA1Y=C1,YB1=D1, andA2X=C2, respectively. As applications, we present necessary and sufficient condition for the previous matrix functionf(X,Y)to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equationsA1Y=C1,YB1=D1,A2X=C2, andB3X+(B3X)*+C3YD3+(C3YD3)*=A3, and give an expression of the general solution to the above-mentioned system when it is solvable.

Download Full-text

Explicit formulas and determinantal representation for η-skew-Hermitian solution to a system of quaternion matrix equations

Filomat ◽

10.2298/fil2008601r ◽

2020 ◽

Vol 34 (8) ◽

pp. 2601-2627

Author(s):

Abdur Rehman ◽

Ivan Kyrchei ◽

Ilyas Ali ◽

Muhammad Akram ◽

Abdul Shakoor

Keyword(s):

Sufficient Conditions ◽

Matrix Equations ◽

Quaternion Matrix ◽

Explicit Formulas ◽

Determinantal Representation ◽

System Of Matrix Equations ◽

Representation Formulas ◽

Special Cases ◽

Hermitian Solution ◽

Necessary And Sufficient

Some necessary and sufficient conditions for the existence of the ?-skew-Hermitian solution quaternion matrix equations the system of matrix equations with ?-skew-Hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X = -X?*; Y=-Y?*, A3XA?*3 + B3YB?*3=C5, are established in this paper by using rank equalities of the coefficient matrices. The general solutions to the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column determinants, we also give determinantal representation formulas of finding their exact solutions that are analogs of Cramer?s rule. A numerical example is also given to demonstrate the main results.

Download Full-text

Parametrized solutions $X$ of the system $AXA = AY A$ and $A^k Y AX = XAY A^k$

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.4051 ◽

2019 ◽

Vol 35 ◽

pp. 503-510 ◽

Cited By ~ 2

Author(s):

David Ferreyra ◽

Marina Lattanzi ◽

Fabián Levis ◽

Néstor Thome

Keyword(s):

General Solution ◽

Matrix Equation ◽

Equation System ◽

Drazin Inverse ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Complex Matrices ◽

The Matrix ◽

Necessary And Sufficient ◽

Parametrized Solutions

Let A and E be n × n given complex matrices. This paper provides a necessary and sufficient condition for the solvability to the matrix equation system given by AXA = AEA and AkEAX = XAEAk, for k being the index of A. In addition, its general solution is derived in terms of a G-Drazin inverse of A. As consequences, new representations are obtained for the set of all G-Drazin inverses; some interesting applications are also derived to show the importance of the obtained formulas.

Download Full-text