An Extension of the Cayley–hamilton Theorem for Nonlinear Time–varying Systems

The classical Cayley-Hamilton theorem (Gantmacher, 1974; Kaczorek, 1988; Lancaster, 1969) says that every square matrix satisfies its own characteristic equation. Let A ∈ Cn×n (the set of n × n complex matrices) and p(s) = det[Ins − A] = ∑n i=0 a si, (an = 1) be the characteristic polynomial of A. Then p(A) = ∑n i=0 aiA i = 0n (the n × n zero matrix). The Cayley Hamilton theorem was extended to rectangular matrices (Kaczorek, 1988; 1995c), block matrices (Kaczorek, 1995b; Victoria, 1982), pairs of commuting matrices (Chang and Chan, 1992; Lewis, 1982; 1986; Kaczorek, 1988), pairs of block matrices (Kaczorek, 1988; 1998) as well as standard and singular two-dimensional linear (2-D) systems (Kaczorek, 1992; 1995a; Smart and Barnett, 1989; Theodoru, 1989). The Cayley-Hamilton theorem and its generalizations were used in control systems, electrical circuits, systems with delays, singular systems, 2-D linear systems, etc., cf. (Busłowicz, 1981; 1982; Kaczorek, 1992; 1994; Lewis, 1982; Mcrtizios and Christodolous, 1986).