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.