Matrix Polynomial Expansion of a Power of a Matrix

Abstract

Any power of an n×n matrix can be expanded by a matrix polynomial of the order n-1, but the coefficients of expansion are not known in closed form. In this paper, it is shown that the coefficients of expansion are given by the solution of a simultaneous equation of the 1st order, whose coefficients compose the Vandermonde matrix. Using the properties of a generalized Vandermonde determinant, coefficients of an expansion of a power of the matrix are obtained in closed form. As the coefficients thus obtained are homogeneous polynomials of eigen-values, and as every term of the polynomial has the same sign, the upper bounds of the absolute values of the coefficients can be obtained easily, if all the eigen-values are located in a disk centered at the origin. If all the eigen-values of a transition matrix of a dynamical system are located in a disc with a radius less than 1 and centered at the origin, the dynamical system is exponentially stable. As the reachable subspace of a dynamical system is spanned by input constraint vectors, multiplied by powers of the transition matrix from the left, the results obtained make a bridge to connect the exponential stability property and the structure of the practically reachable subspace of a dynamical system.