M-dimensional Cayley-Hamilton theorem

The theorem states that every block square matrix satisfies its own m-D (m-dimensional, m⩾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 N₁ . . ., N _m, such that N₁+ . . . +N_m ⩽M