How to compute the elementary divisors of the tensor product of two matrices

Abstract

Let A and B be square matrices over a field F of characteristic p. Suppose A and B each have a single elementary divisor (X−λ) ^m and (X−μ) ⁿ , respectively, where 0≠λ, μεF. By studying the representation algebra of a cyclic group whose order is a power of p, we give a procedure for computing the elementary divisors of A⨷B. For example, if p=2m=13 and n=11 the appropriate group has order 16 and the algebra has a basis ν₁…,ν₁₆. By computing ν₁₃ν₁₁=2ν₄+ν₇+8ν₁₆ we deduce that the elementary divisors of A⨷B are (X−λμ)⁴, twice, (X−λμ)⁷, and (X−λμ)¹⁶, 8 times.