ABSTRACT
The measure of the magnitude of A, x, or b is called its norm and denoted by IIAII, Ilxll, and IIbll, respectively. Norms have the following properties:
(1.17Ia) (1.17Ib) (1.17Ic) (1.17Id) (1.17Ie)
The norm of a scalar is its absolute value. Thus, IIkll = Ikl. There are several definitions of the norm of a vector. Thus,
IIxll oo = max Ixdl~i~/1
Sum of magnitudes
Euclidean norm Maximum magnitude norm
(1.172a) (I.I72b) (1.172c)
The Euclidean norm is the length of the vector in n-space. In a similar manner, there are several definitions of the norm of a matrix. Thus,
IIAlie = ~)~ af) 1.6.3.2. Condition Number
Maximum column sum
Maximum row sum
Spectral norm
Euclidean norm
(1.173a)
(1.173b)
(1.173c)
(1.173d)
The condition number of a system is a measure of the sensitivity of the system to small changes in any of its elements. Consider a system of linear algebraic equations:
Ax=b For Eq. (1.174),
(1.174)
(1.175) Consider a slightly modified form of Eq. (1.174) in which b is altered by c5b, which causes a change in the solution c5x. Thus,
A(x + c5x) = b + c5b Subtracting Eq. (1.174) from Eq. (1.176) gives
A c5x = c5b Solving Eq. (1.177) for c5x gives
(1.176)
(1.177)
(1.178)
