Characterization for the general solution to a system of matrix equations with quadruple variables

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.

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

Algebra Colloquium ◽

10.1142/s1005386716000195 ◽

2016 ◽

Vol 23 (01) ◽

pp. 167-180 ◽

Cited By ~ 1

Author(s):

Guangjing Song ◽

Shaowen Yu ◽

Ming Chen

Keyword(s):

General Solution ◽

Sufficient Condition ◽

Matrix Equations ◽

Necessary And Sufficient Condition ◽

System Of Matrix Equations ◽

Hermitian Solution ◽

Necessary And Sufficient

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.

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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].

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The{P,Q,k+1}-Reflexive Solution to System of Matrix EquationsAX=C,XB=D

Mathematical Problems in Engineering ◽

10.1155/2015/464385 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Chang-Zhou Dong ◽

Qing-Wen Wang

Keyword(s):

General Solution ◽

Least Squares ◽

Approximation Problem ◽

Optimal Approximation ◽

Matrix Equations ◽

Solvability Conditions ◽

Least Squares Solution ◽

System Of Matrix Equations ◽

Conjugate Transpose

LetP∈Cm×mandQ∈Cn×nbe Hermitian and{k+1}-potent matrices; that is,Pk+1=P=P⁎andQk+1=Q=Q⁎,where·⁎stands for the conjugate transpose of a matrix. A matrixX∈Cm×nis called{P,Q,k+1}-reflexive (antireflexive) ifPXQ=X (PXQ=-X). In this paper, the system of matrix equationsAX=CandXB=Dsubject to{P,Q,k+1}-reflexive and antireflexive constraints is studied by converting into two simpler cases:k=1andk=2.We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered.

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Solvability Conditions and General Solution of a System of Matrix Equations Involving η-Skew-Hermitian Quaternion Matrices

Symmetry ◽

10.3390/sym13101825 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1825

Author(s):

Abdur Rehman ◽

Israr Ali Khan ◽

Rukhshanda Anjum ◽

Iftikhar Hussain

Keyword(s):

General Solution ◽

Matrix Equations ◽

Solvability Conditions ◽

Numerical Example ◽

Quaternion Matrices ◽

System Of Matrix Equations ◽

Special Cases

In this article, we study the solvability conditions and the general solution of a system of matrix equations involving η-skew-Hermitian quaternion matrices. Several special cases of this system are discussed, and we recover some well-known results in the literature. An algorithm and a numerical example for the validation of our main result are also provided.

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