Triangularizing matrix polynomials

For an algebraically closed field F, we show that any matrix polynomial P(λ)∈F[λ]n×m, n⩽m, can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors. We also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes 1×1 and 2×2. The proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors.