Efficient numerical method for the discrete-time symmetric matrix polynomial equation

1998 ◽  
Vol 145 (5) ◽  
pp. 443-448 ◽  
Author(s):  
D. Henrion ◽  
M. Šebek
Keyword(s):  
Numerical Method ◽  
Discrete Time ◽  
Matrix Polynomial ◽  
Symmetric Matrix ◽  
Polynomial Equation ◽  
Matrix Polynomial Equation

scholarly journals The structure of solutions of the matrix linear unilateral polynomial equation with two variables

2017 ◽  
Vol 9 (1) ◽  
pp. 48-56
Author(s):  
N.S. Dzhaliuk ◽  
V.M. Petrychkovych
Keyword(s):  
Matrix Polynomial ◽  
Sufficient Conditions ◽  
Polynomial Equation ◽  
Minimal Degree ◽  
Structure Of Solutions ◽  
Arbitrary Matrix ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Matrix Polynomial Equation ◽  
Necessary And Sufficient

We investigate the structure of solutions of the matrix linear polynomial equation $A(\lambda)X(\lambda)+B(\lambda)Y(\lambda)=C(\lambda),$ in particular, possible degrees of the solutions. The solving of this equation is reduced to the solving of the equivalent matrix polynomial equation with matrix coefficients in triangular forms with invariant factors on the main diagonals, to which the matrices $A (\lambda), B(\lambda)$ \ and \ $C(\lambda)$ are reduced by means of semiscalar equivalent transformations. On the basis of it, we have pointed out the bounds of the degrees of the matrix polynomial equation solutions. Necessary and sufficient conditions for the uniqueness of a solution with a minimal degree are established. An effective method for constructing minimal degree solutions of the equations is suggested. In this article, unlike well-known results about the estimations of the degrees of the solutions of the matrix polynomial equations in which both matrix coefficients are regular or at least one of them is regular, we have considered the case when the matrix polynomial equation has arbitrary matrix coefficients $A(\lambda)$ and $B(\lambda).$ 

On solutions of second-order matrix polynomial equation of high degree

2021 ◽  
pp. 2250100
Author(s):  
Lu Tan ◽  
Xue-Han Cheng ◽  
Tong-Song Jiang ◽  
Si-Tao Ling
Keyword(s):  
Matrix Polynomial ◽  
Sufficient Conditions ◽  
Polynomial Equation ◽  
Second Order ◽  
General Solutions ◽  
Order Matrix ◽  
Algebraic Properties ◽  
Analytic Expressions ◽  
Matrix Polynomial Equation ◽  
High Degree

In this paper, we focus on discussing diagonal solutions and general solutions of second-order matrix polynomial equation of high degree in complex field. By characterizing some algebraic properties of the mentioned two types of the solutions, we present sufficient conditions that a general second-order matrix polynomial equation has diagonal solutions or general solutions. Analytic expressions of the solutions, as well as the corresponding algorithms for finding the solutions are provided. An example is given so as to verify the theoretical results we have derived.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Yunbo Tian ◽  
Chao Xia
Keyword(s):  
Matrix Equation ◽  
Matrix Polynomial ◽  
Polynomial Equation ◽  
Sylvester Matrix Equation ◽  
Sylvester Matrix ◽  
Low Degree ◽  
Matrix Polynomial Equation ◽  
Degree Conditions

We study the low-degree solution of the Sylvester matrix equation A 1 λ + A 0 X λ + Y λ B 1 λ + B 0 = C 0 , where A 1 λ + A 0 and B 1 λ + B 0 are regular. Using the substitution of parameter variables λ , we assume that the matrices A 0 and B 0 are invertible. Thus, we prove that if the equation is solvable, then it has a low-degree solution L λ , M λ , satisfying the degree conditions δ L λ < Ind A 0 − 1 A 1  and  δ M λ < Ind B 1 B 0 − 1 .

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