Nonlinear Control Sy.. - Free

48 CHAPTER 2. MATHEMATICAL PRELIMINARIES
2.7.1 Bounded Linear Operators and Matrix Norms
Now consider a function L mapping vector spaces X into Y.
Definition 2.19 A function L : X --4 Y is said to be a linear operator (or a linear map,
or a linear transformation) if and only if given any X1, X2 E X and any A, p E R
L(Axi + µx2) = .L(x1) + pL(x2). (2.13)
The function L is said to be a bounded linear operator if there exist a constant M such that
JIL(x)Ij < MIjxII Vx E X. (2.14)
The constant M defined in (2.14) is called the operator norm. A special case of
interest is that when the vector spaces X and Y are Rn and 1R', respectively. In this case
all linear functions A : Rn -* R'n are of the form
y=Ax, xER', yERm
where A is a m x n of real elements. The operator norm applied to this case originates a
matrix norm. Indeed, given A E II8"`n, this matrix defines a linear mapping A : Rn -* R'n
of the form y = Ax. For this mapping, we define
def IIAxp =
zoo IIx11P 1121 1
(2.15)
Where all the norms on the right-hand side of (2.15) are vector norms. This norm is
sometimes called the induced norm because it is "induced" by the p vector norm. Important
special cases are p = 1, 2, and oo. It is not difficult to show that
IIAIIc
m
max IAxjIi = maxE lazal (2.16)
IlxIIi=1 7
x=1
max IIAxII2 =
IIXI12=1
Amax(ATA) (2.17)
n
max IIAxII. = maxE lain (2.18)
Ilxlloo=1
where Amax (AT A) represents the maximum eigenvalue of AT A.
i
J=1