Nonlinear Control Sy.. - Free

40 CHAPTER 2. MATHEMATICAL PRELIMINARIES
Definition 2.9 The null space of a linear function A : X --+ Y zs the set N(A), defined by
N(A)={xEX: Ax=0}
It is straightforward to show that N(A) is a vector space. The dimension of this
vector space, denoted dimA((A), is important. We now state the following theorem without
proof.
Theorem 2.3 Let A E .Pn". Then A has the following property:
rank(A) + dimM(A) = n.
2.4.1 Eigenvalues, Eigenvectors, and Diagonal Forms
Definition 2.10 Consider a matrix A E j,,xn. A scalar A E F is said to be an eigenvalue
and a nonzero vector x is an eigenvector of A associated with this eigenvalue if
or
Ax = Ax
(A - AI)x = 0
thus x is an eigenvector associated with A if and only if x is in the null space of (A - AI).
Eigenvalues and eigenvectors are fundamental in matrix theory and have numerous
applications. We first analyze their use in the most elementary form of diagonalization.
Theorem 2.4 I f A E R... with eigenvalues Al, A 2 ,--', An has n linearly independent
eigenvectors vl, V2i , vn, then it can be expressed in the form
where D = diag{Al, A2i . . . , An}, and
A = SDS-1
S= vl ... vn
L