On the calculation of matrix polynomials

The purpose of this note is to prove a theorem which permits the calculation of the vector b without the necessity of transforming the matrix A into the Jordan form for the Cayley-Hamilton theorem. The only requirement is the calculation of the characteristic polynomial of the matrix A, which may be easily done using the existing computational procedures. The author includes a computer program listing of a subroutine that calculates the coefficients of the n-dimensional vector b. This subroutine makes use of certain standard IBM subroutines, as SMPY, GMPRD, GMSUB, and GMADD, from the IBM 1130 Scitific Subroutine Package, and utilized another subroutine CHEQ described by the author in ibid., vol AC-20, pp. 134-135, Feb. 1975, that calculates the coefficients of the characteristic polynomial of the matrix A.