M-dimensional Cayley-Hamilton theorem

Abstract

The theorem states that every block square matrix satisfies its own m-D (m-dimensional, m>or=1) matrix characteristic polynomial. The exact statement and a simple proof of this theorem are given. The theorem refers to a matrix A subdivided into m blocks, and hence having dimension at least m. The conclusion is that every square matrix A with dimension M satisfies several m-D characteristic matrix polynomials with degrees N/sub 1/ . . ., N/sub m/, such that N/sub 1/+ . . . +N/sub m/<or=M.