General Theory of Linear Systems

Abstract

The main topic of this chapter is the structure of solutions of a linear system $$ \frac{{d\vec y}}{{dt}} = A(t)\vec y + \vec b(t), $$ (LP) where entries of the n × n matrix A(t) are complex-valued (i.e.,ℂ-valued) continuous functions of a real independent variable t, and the ℂn-valued function \(\mathop{{{\text{ }}b}}\limits^{ \to } (t)\) is continuous in t. The existence and uniqueness of solutions of problem (LP) were given by Theorem I-3-5. In §IV-1, we explain some basic results concerning n x n matrices whose entries are complex numbers. In particular, we explain the S-N decomposition (or the Jordan-Chevalley decomposition) of a matrix (cf. Definition IV-1-12; also see [Bou, Chapter 7], [HirS, Chapter 6], and [Hum, pp. 17-18]). The S-N decomposition is equivalent to the block-diagonalization which separates distinct eigenvalues. It is simpler than the Jordan canonical form. The basic tools for achieving this decomposition are the Cayley-Hamilton theorem (cf. Theorem IV-1-5) and the partial fraction decomposition of reciprocal of the characteristic polynomial. It is relatively easy to obtain this decomposition with an elementary calculation if all eigenvalues of a given matrix are known (cf. Examples IV-1-18 and IV-1-19). In §IV-2, we explain the general aspect of linear homogeneous systems. Homogeneous systems with constant coefficients are treated in §IV-3. More precisely speaking, we define etA and discuss its properties. In §IV-4, we explain the structure of solutions of a homogeneous system with periodic coefficients. The main result is the Floquet theorem (cf. Theorem IV-4-1 and [F1]).