On the Low-Degree Solution of the Sylvester Matrix Polynomial Equation

Tian, Yunbo; Xia, Chao

doi:10.1155/2021/4612177

Cited by 6 publications

(2 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the matrix equation AX � B, the Hermitian reflexive, antireflexive, generalized reflexive, and antireflexive solutions have been discussed in [15,16]. And different solution sets for other equations have also been discussed in [17][18][19][20]. Besides, the optimization problems of different matrices have also been studied in [21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

Dong

2022

Journal of Mathematics

View full text Add to dashboard Cite

This paper involves related inverse eigenvalue problem and least-squares problem of skew-Hermitian {P,k + 1}-reflexive(antireflexive) matrices and their optimal approximation problems. The above problems are studied by converting them into two simpler cases: k = 1 and k = 2. Firstly, with some special properties of skew-Hermitian {P,k + 1}-reflexive(antireflexive) matrices, the necessary and sufficient conditions for the solvability and the general solution are presented, and the solution of corresponding optimal approximation problems also given, respectively. Then, we give the least-squares solution of AX = B satisfying the special condition by the singular value decomposition. Finally, we give an algorithm and an example to illustrate our results.

show abstract

Section: Introductionmentioning

confidence: 99%

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

Dong

2022

Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…In the many problems, in addition to the solvability of such matrix equations and the finding of general solutions, solutions with certain properties are needed. The solutions of bounded degrees to the linear matrix polynomial equations are indicated in [5][6][7][8][9]. In [10], integer solutions of such matrix equations over quadratic rings are found.…”

mentioning

confidence: 99%

Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations

Dzhaliuk,

Petrychkovych

2024

Mat. Stud.

View full text Add to dashboard Cite

We investigate the solutions of the Sylvester-type matrix polynomial equation $$A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda),$$ where\ $A(\lambda),$ \ $ B(\lambda),$\ and \ $C(\lambda)$ are the polynomial matrices with elements in a ring of polynomials \ $\mathcal{F}[\lambda],$ \ $\mathcal{F}$ is a field,\ $X(\lambda)$\ and \ $Y(\lambda)$ \ are unknown polynomial matrices. Solving such a matrix equation is reduced to the solving a system of linear equations $$G \left\|\begin{array}{c}\mathbf{x} \\ \mathbf{y} \end{array} \right\|=\mathbf{c}$$ over a field $\mathcal{F}.$ In this case, the Kronecker product of matrices is applied. In terms of the ranks of matrices over a field $\mathcal{F},$ which are constructed by the coefficients of the Sylvester-type matrix polynomial equation,the necessary and sufficient conditions for the existence of solutions \ $X_0(\lambda)$\ and \ $Y_0(\lambda)$ \ of given degrees to the Sylvester-type matrix polynomial equation are established. The solutions of this matrix polynomial equation are constructed from the solutions of the linear equations system.As a consequence of the obtained results, we give the necessary and sufficient conditions for the existence of the scalar solutions \ $X_0$\ and \ $Y_0,$ \ whose entries are elements in a field $\mathcal{F},$ to the Sylvester-type matrix polynomial equation.

show abstract

Matrix Linear Bilateral Equations Over Different Domains, Methods for the Construction of Solutions, and Description of Their Structure

Dzhaliuk,

Petrychkovych

2024

J Math Sci

View full text Add to dashboard Cite

On the Low-Degree Solution of the Sylvester Matrix Polynomial Equation

Abstract: We study the low-degree solution of the Sylvester matrix equation A 1 λ + … Show more

Cited by 6 publications

References 19 publications

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations

Matrix Linear Bilateral Equations Over Different Domains, Methods for the Construction of Solutions, and Description of Their Structure

Contact Info

Product

Resources

About