On 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 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.