On a quadratic matrix equation associated with an M-matrix∗
نویسنده
چکیده
We study the quadratic matrix equation X2 − EX − F = 0, where E is diagonal and F is an M -matrix. Quadratic matrix equations of this type arise in noisy Wiener–Hopf problems for Markov chains. The solution of practical interest is a particular M -matrix solution. The existence and uniqueness of M -matrix solutions and numerical methods for finding the desired M -matrix solution are discussed by transforming the equation into an equation that belongs to a special class of nonsymmetric algebraic Riccati equations (AREs). We also discuss the general nonsymmetric ARE and describe how we can use the Schur method to find the stabilizing or almost stabilizing solution if it exists. The desired M -matrix solution of the quadratic matrix equation (a special nonsymmetric ARE by itself) turns out to be the unique stabilizing or almost stabilizing solution.
منابع مشابه
Analytical aspects of the interval unilateral quadratic matrix equations and their united solution sets
This paper introduces the emph{interval unilateral quadratic matrix equation}, $IUQe$ and attempts to find various analytical results on its AE-solution sets in which $A,B$ and $CCC$ are known real interval matrices, while $X$ is an unknown matrix. These results are derived from a generalization of some results of Shary. We also give sufficient conditions for non-emptiness of some quasi-solutio...
متن کاملOn the solving matrix equations by using the spectral representation
The purpose of this paper is to solve two types of Lyapunov equations and quadratic matrix equations by using the spectral representation. We focus on solving Lyapunov equations $AX+XA^*=C$ and $AX+XA^{T}=-bb^{T}$ for $A, X in mathbb{C}^{n times n}$ and $b in mathbb{C} ^{n times s}$ with $s < n$, which $X$ is unknown matrix. Also, we suggest the new method for solving quadratic matri...
متن کاملOn the square root of quadratic matrices
Here we present a new approach to calculating the square root of a quadratic matrix. Actually, the purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine an explicit formula for all the square roots of $2times 2$ matrices.
متن کاملInvolutiveness of linear combinations of a quadratic or tripotent matrix and an arbitrary matrix
In this article, we characterize the involutiveness of the linear combination of the forma1A1 +a2A2 when a1, a2 are nonzero complex numbers, A1 is a quadratic or tripotent matrix,and A2 is arbitrary, under certain properties imposed on A1 and A2.
متن کاملThe use of inverse quadratic radial basis functions for the solution of an inverse heat problem
In this paper, a numerical procedure for an inverse problem of simultaneously determining an unknown coefficient in a semilinear parabolic equation subject to the specification of the solution at an internal point along with the usual initial boundary conditions is considered. The method consists of expanding the required approximate solution as the elements of the inverse quadrati...
متن کامل