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

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 .

On solutions of second-order matrix polynomial equation of 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.

Convergence of a modified Newton method for a matrix polynomial equation arising in stochastic problem

The Newton iteration is considered for a matrix polynomial equation which arises in stochastic problem. In this paper, it is shown that the elementwise minimal nonnegative solution of the matrix polynomial equation can be obtained using Newton's method if the equation satisfies the sufficient condition, and the convergence rate of the iteration is quadratic if the solution is simple. Moreover, it is shown that the convergence rate is at least linear if the solution is non-simple, but a modified Newton method whose iteration number is less than the pure Newton iteration number can be applied. Finally, numerical experiments are given to compare the effectiveness of the modified Newton method and the standard Newton method.

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

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 $B(\lambda).$