On the solving matrix equations by using the spectral representation
Subject Areas : Linear and multilinear algebra; matrix theory
A. M. Nazari
1
(Department of Mathematics, Faculty of Science, Arak university, Arak, PO. Box 38156-8-8349, Iran)
S. Mollaghasemi
2
(Department of Mathematics, Faculty of Science, Arak university, Arak, PO. Box 38156-8-8349, Iran)
F. Bahmani
3
(Department of Mathematics, Faculty of Science, Arak university, Arak, PO. Box 38156-8-8349, Iran)
Keywords: Lyapanov equation, square matrix equations, spectral representation,
Abstract :
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 matrix equations $AX^{2}+BX+C=0$, where $A, B, C, X \in \mathbb{C}^{n \times n}$ and $X$ is unknown matrix with similar method.
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