Abstract. One of the interesting nonlinear matrix equations is the quadratic matrix equation defined bywhere X is a n × n unknown real matrix, and A, B and C are n × n given matrices with real elements. Another one is the matrix polynomialNewton'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 Fréchet derivative. Finally, we give some numerical examples.

Abstract. In this paper a nonlinear matrix equation is considered which has the formwhere X is an n × n unknown real matrix and A m , A m−1 , . . . , A 0 are n × n matrices with real elements. Newtons method is applied to find the skew-symmetric solvent of the matrix polynomial P (X). We also suggest an algorithm which converges the skew-symmetric solvent even if the Fréchet derivative of P (X) is singular.

Abstract. In this paper we consider the quadratic matrix equation which can be defined bywhere X is a n × n unknown real matrix; A, B and C are n × n given matrices with real elements. Newton's method is considered to find the skew-symmetric solvent of the nonlinear matrix equations Q(X). We also show that the method converges the skew-symmetric solvent even if the Fréchet derivative is singular. Finally, we give some numerical examples.