2.1 Introduction

Results from matrix theory and linear algebra play an important role in the theory of finite dimensional control systems. This chapter presents key results from matrix theory that are essential for understanding control theory concepts.

In Section 2.2, linear dependence and independence of vectors in ℝn and basis and dimension of a subspace are discussed. In Section 2.3, elementary row operation, row reduced echelon form, range and rank of a matrix, and inverse and determinant of a square matrix are presented. Section 2.4 is on solving a system of linear algebraic equations. In Section 2.5, null space and nullity are defined. Eigenvalues and eigenvectors of a matrix and related results are discussed in Section 2.6. Canonical forms (diagonal and Jordan forms) are explained in Section 2.7. Function of a matrix and computation of exponential of a matrix are presented in Section 2.8. In Section 2.9, special matrices and their properties are discussed.

2.2 The ℝn and ℂn Spaces

Let ℝ and ℂ be the set of all real and complex numbers, respectively. The ndimensional Euclidean space, denoted by ℝn, is defined as ℝn = {(x1,x2, . . . ,xn) : xi ∈ ℝ : i = 1,2, . . . ,n}. Each element of ℝn can be considered as a vector with ncomponents, and Rn is called the n-dimensional vector space over ℝ (see Definition 2.2.12).