Smith form and Kronecker canonical form

Abstract

This chapter treats matrix polynomials with quaternion coefficients. A diagonal form (known as the Smith form), which asserts that every quaternion matrix polynomial can be brought to a diagonal form under pre- and postmultiplication by unimodular matrix polynomials, is proved for such polynomials. In contrast to matrix polynomials with real or complex coefficients, a Smith form is generally not unique. For matrix polynomials of first degree, a Kronecker form—the canonical form under strict equivalence—is available, which this chapter presents with a complete proof. Furthermore, the chapter gives a comparison for the Kronecker forms of complex or real matrix polynomials with the Kronecker forms of such matrix polynomials under strict equivalence using quaternion matrices.