A Note on Weak and Strong Linearizations of Regular Matrix Polynomials

A paper of Tan and Pugh [TP] raises the question of ambiguity in a frequently used form of linearization when applied to regular matrix polynomials. Here, further insight into this question is provided, as well as a reminder of a stronger form of linearization (for which ambiguities are removed) introduced by Gohberg et al. [GKL]. Let A0, A1, . . . , An ∈ Cn×n, and define the matrix polynomial L(λ) = ∑l i=0 λ Ai. Then L(λ) is said to be regular if det L(λ) is not identically equal to zero. We consider only regular matrix polynomials, and notice that Al = 0 is admitted. The degree of L(λ) is the largest j for which Aj 6= 0. Thus, it may happen that l > deg(L). Some important ideas for this discussion are as follows: Two regular matrix polynomials A(λ), B(λ) of the same size are said to be equivalent if there are unimodular matrix polynomials E(λ), F (λ) such that A(λ) = E(λ)B(λ)F (λ). The canonical form under equivalence is the well-known Smith form, and it reveals the structure of the invariant polynomials and (finite) elementary divisors. There is also a local Smith form in which the transforming matrices E(λ) and F (λ) are invertible near an eigenvalue λ0 and the elementary divisor structure of the single eigenvalue λ0 is revealed (see [BGR], for example). i.e. invertible with non-vanishing determinant independent of λ