Spectral structures of the generalized companion form and applications

Linearization of regular matrix polynomials

“…This is all very natural, but a problem appears: Although the (total) algebraic multiplicities of the infinite eigenvalue of P and the zero eigenvalue of P # agree, the partial multiplicities of the eigenvalue at infinity depend on the choice of the linear pencil λA − B used in (1.2). This was noticed in [14] for example, and was further investigated in [10] where the following result appears.…”

“…See the notes on Theorem 2 below. More recently, this idea has become a standard tool in the theory of matrices and operators (as in [1] and [13]), and systems theory (see [9] and [14], for example).…”

On the computation of the Jordan canonical form of regular matrix polynomials

“…Proof By (5) and [10, Theorem 7.7.1] (see also [13,Proposition 2]), it is clear that J P,F = J L P ,F . For the infinite spectrum, consider the algebraic dual matrix polynomialP (λ) in (3) and its companion linearization LP (λ).…”

Linearizations of polynomial matrices with symmetries and their applications

“…An immediate consequence of the above relation is that the first companion form has the same finite elementary divisors structure with T (s). However, in [13], [10], this ELA 108 E.N. Antoniou and S. Vologiannidis important property of the first companion form of T (s) has been shown to hold also for the infinite elementary divisors structures of P (s) and T (s).…”