Journal of the
Korean Mathematical Society JKMS

One of the interesting nonlinear matrix equations is the quadratic matrix equation defined by \begin{equation*} Q(X)=AX^2+BX+C=0, \end{equation*} where $X$ is a $n\times n$ unknown real matrix, and $A,B$ and $C$ are $n\times n$ given matrices with real elements. Another one is the matrix polynomial \begin{equation*} P(X)=A_0X^m+A_1X^{m-1}+\cdots+A_m=0, \quad X, A_i\in\mathbb{R}^{n\times n}. \end{equation*} Newton's method is used to find the symmetric and bisymmetric solvents of the nonlinear matrix equations $Q(X)$ and $P(X)$. The method does not depend on the singularity of the Fr\'{e}chet derivative. Finally, we give some numerical examples.

Keywords: quadratic matrix equation, matrix polynomial, solvent, Newton's method, iterative algorithm, symmetric, bisymmetric

MSC numbers: Primary 65F30, 65H10