Extreme Ranks of a Partial Banded Block Quaternion Matrix Expression Subject to Some Matrix Equations with Applications

2011 ◽  
Vol 18 (02) ◽  
pp. 333-346 ◽  
Author(s):  
Dezhong Lian ◽  
Qingwen Wang ◽  
Yan Tang
Keyword(s):  
Quaternion Algebra ◽  
Quadratic System ◽  
Block Matrix ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Necessary And Sufficient Condition ◽  
System A ◽  
Matrix Formula ◽  
Matrix Expression ◽  
Necessary And Sufficient

We establish the formulas of the maximal and minimal ranks of a 3 × 3 partial banded block matrix [Formula: see text] where X and Y are a pair variant quaternion matrices subject to linear quaternion matrix equations A1X=C1, XB1=C2, A2Y=D1, YB2=D2. As applications, we present a necessary and sufficient condition for the solvability to the quadratic system A1X=C1, XB1=C2, A2Y=D1, YB2=D2, XPY=J over the quaternion algebra. We also give the conditions for the rank invariance of the quadratic matrix expression XPY=J subject to the linear quaternion matrix equations mentioned above.

scholarly journals Ranks of a Constrained Hermitian Matrix Expression with Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
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.

scholarly journals Some quaternion matrix equations involving Φ-Hermicity

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5097-5112 ◽  
Author(s):  
Zhuo-Heng He
Keyword(s):  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Matrix Decomposition ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Existence Of A Solution ◽  
Quaternion Matrices ◽  
Real Quaternion ◽  
Necessary And Sufficient ◽  
The Given

Let H be the real quaternion algebra and Hmxn denote the set of all m x n matrices over H. For A ? Hm x n, we denote by A? the n x m matrix obtained by applying ? entrywise to the transposed matrix At, where ? is a nonstandard involution of H. A ? Hnxn is said to be ?-Hermitian if A = A?. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number (A,B,C,D), where A is ?-Hermitian, and B,C,D are general matrices. Using this simultaneous matrix decomposition, we derive necessary and sufficient conditions for the existence of a solution to some real quaternion matrix equations involving ?-Hermicity in terms of ranks of the given real quaternion matrices. We also present the general solutions to these real quaternion matrix equations when they are solvable. Finally some numerical examples are presented to illustrate the results of this paper.

A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations

2017 ◽  
Vol 24 (01) ◽  
pp. 169-180 ◽  
Author(s):  
Zhuoheng He ◽  
Qingwen Wang
Keyword(s):  
General Solution ◽  
Discrete Time ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Necessary And Sufficient ◽  
Sylvester Matrix Equations

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations [Formula: see text] over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
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.

Finding Cut-Edges and the Minimum Spanning Tree via Semi-Tensor Product Approach

2018 ◽  
Vol 6 (5) ◽  
pp. 459-472
Author(s):  
Xujiao Fan ◽  
Yong Xu ◽  
Xue Su ◽  
Jinhuan Wang
Keyword(s):  
Tensor Product ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Product Approach ◽  
Matrix Expression ◽  
Necessary And Sufficient ◽  
Logical Functions

Abstract Using the semi-tensor product of matrices, this paper investigates cycles of graphs with application to cut-edges and the minimum spanning tree, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for cycles of graph, based on which a new necessary and sufficient condition is established to find all cycles for any graph. Secondly, using the necessary and sufficient condition of cycles, two algorithms are established to find all cut-edges and the minimum spanning tree, respectively. Finally, the study of an illustrative example shows that the results/algorithms presented in this paper are effective.

scholarly journals Pure PSVD approach to Sylvester-type quaternion matrix equations

2019 ◽  
Vol 35 ◽  
pp. 266-284 ◽  
Author(s):  
Zhuo-Heng He
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Singular Value ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Existence Of A Solution ◽  
Quaternion Matrices ◽  
Pure Product ◽  
Value Decomposition ◽  
Necessary And Sufficient

In this paper, the pure product singular value decomposition (PSVD) for four quaternion matrices is given. The system of coupled Sylvester-type quaternion matrix equations with five unknowns $X_{i}A_{i}-B_{i}X_{i+1}=C_{i}$ is considered by using the PSVD approach, where $A_{i},B_{i},$ and $C_{i}$ are given quaternion matrices of compatible sizes $(i=1,2,3,4)$. Some necessary and sufficient conditions for the existence of a solution to this system are derived. Moreover, the general solution to this system is presented when it is solvable.

scholarly journals Completing a2×2Block Matrix of Real Quaternions with a Partial Specified Inverse

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yong Lin ◽  
Qing-Wen Wang
Keyword(s):  
General Expression ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Block Matrix ◽  
The Real ◽  
Real Quaternion ◽  
Completion Problem ◽  
Left Block ◽  
Necessary And Sufficient ◽  
Real Quaternions

This paper considers a completion problem of a nonsingular2×2block matrix over the real quaternion algebraℍ: Letm1,  m2,  n1,  n2be nonnegative integers,m1+m2=n1+n2=n>0, andA12∈ℍm1×n2, A21∈ℍm2×n1, A22∈ℍm2×n2, B11∈ℍn1×m1be given. We determine necessary and sufficient conditions so that there exists a variant block entry matrixA11∈ℍm1×n1such thatA=(A11A12A21A22)∈ℍn×nis nonsingular, andB11is the upper left block of a partitioning ofA-1. The general expression forA11is also obtained. Finally, a numerical example is presented to verify the theoretical findings.

scholarly journals Stability Analysis ofθ-Methods for Neutral Multidelay Integrodifferential System

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Y. Xu ◽  
J. J. Zhao ◽  
Z. N. Sui
Keyword(s):  
Numerical Stability ◽  
Numerical Experiments ◽  
Analytic Solutions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Delay ◽  
Integrodifferential System ◽  
System A ◽  
The Stability ◽  
Necessary And Sufficient

This paper studies the stability of a class of neutral delay integrodifferential system. A necessary and sufficient condition of stability for its analytic solutions is considered. The improvedθ-methods are developed. Some numerical stability properties are obtained and numerical experiments are given.

Extreme Ranks of Real Matrices in Solution of the Quaternion Matrix Equation AXB = C with Applications

2010 ◽  
Vol 17 (02) ◽  
pp. 345-360 ◽  
Author(s):  
Qingwen Wang ◽  
Shaowen Yu ◽  
Wei Xie
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Real Solution ◽  
Quaternion Matrix ◽  
Necessary And Sufficient Condition ◽  
Solvability Conditions ◽  
Quaternion Matrix Equation ◽  
Necessary And Sufficient ◽  
Real Matrices

In this paper, for a consistent quaternion matrix equation AXB = C, the formulas are established for maximal and minimal ranks of real matrices X1, X2, X3, X4 in solution X = X1 + X2i + X3j + X4k. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices E, F, G, H in a generalized inverse (A +Bi + Cj + Dk)- = E + Fi + Gj + Hk of a quaternion matrix A + Bi + Cj + Dk are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations A1XB1 = C1 and A2XB2 = C2 to have a common real solution.

scholarly journals On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems

2021 ◽  
Vol 11 (2) ◽  
pp. 596
Author(s):  
Mei Zhang ◽  
Boutaïeb Dahhou ◽  
Ze-tao Li
Keyword(s):  
Local Level ◽  
Sufficient Condition ◽  
Global Level ◽  
Necessary And Sufficient Condition ◽  
Initial State ◽  
Interconnected System ◽  
Global System ◽  
System A ◽  
Necessary And Sufficient ◽  
Local Input

In this paper, the invertibility of an interconnected system that consists of two dynamic subsystems was studied. It can be viewed as the distinguishability of the impacts of local input on the final global output, that is to say, whether the input at the local level can be recovered uniquely under a given output at the global level and initial state. The interconnected system constitutes two dynamic subsystems connected in a cascade manner. In order to guarantee the invertibility of the studied system, a necessary and sufficient condition was established. On the condition that both individual subsystems are invertible, the invertibility of the global system can be guaranteed. In order to recover the local input which generates a given global output, an algorithm was proposed for the studied interconnected system. Numerical examples were considered to confirm the effectiveness and robustness of the proposed algorithm.

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