Matrix sign function in the problems of analysis and design of the linear systems

The properties of the matrix sign function, a relatively new object of the matrix theory, were described. In the modern computational algebra, it underlies an efficient technology enabling one to resolve the topical problems of the control theory. The potentialities of this technology were demonstrated by the example of the spectral problems and the Lyapunov, Sylvester, and Riccati algebraic matrix equations. Practical criteria for stability of the linear dynamic system were proposed on the basis of the iterative procedure for computation of the matrix sign function. An example of stability analysis of the mathematical model of a high-order power system was presented. A generalized matrix sign function for the generalized bundles of square matrices was described. For the generalized bundles of rectangular matrices, the matrix sign function was defined relying on the algebra of linear relations.