In this article the criterion for convergence of a sequence of powers to zero matrix and its practical applications are presented. The linear discrete dynamic system with constant coefficients is explored and the sufficient and necessary condition for the all eigenvalues of a given matrix have a negative real part in the complex plane is obtained. It is proved that if the maximum of modules of the all eigenvalues of the Jacobi matrix at zero of a complex valued holomorphic vector function with several variables is less than one then the zero fixed point of the function is attractive.