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Ax = b be a nonsingular linear system, and let Ax = b be a perturbed system.
−1 (b − b) . In order to isolate the
The normwise absolute error is x − x = A
perturbation and derive a bound of the form A
b − b , we have to deﬁne a
norm for matrices.
Deﬁnition 2.12. A matrix norm
properties: · is a function from Cm×n to R with three N1: A ≥ 0 for all A ∈ Cn×m , and A = 0 if and only if A = 0. N2: A + B ≤ A + B for all A, B ∈ Cm×n (triangle inequality). N3: α A = |α | A for all α ∈ C, A ∈ Cm×n . Because of the triangle inequality, matrix norms are well-conditioned, in the
absolute sense and in the relative sense.
Fact 2.13. If A, E ∈ Cm×n , then A+E − A ≤ E. Proof.
The triangle inequality implies A + E ≤ A + E , hence
A + E − A ≤ E . Similarly A = (A + E) − E ≤ A + E + E ,
so that − E ≤ A + E − A . The result follows from
− E ≤ A+E − A ≤ E .
The matrix p-norms below are based on the vector p-norms and measure
how much a matrix can stretch a unit-norm vector. Downloaded 01/17/14 to 188.8.131.52. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 34 2. Sensitivity, Errors, and Norms Fact 2.14 (Matrix p-Norms). Let A ∈ Cn×m . The p-norm
A p Ax p
= max Ax
x p =1
xp = max
x =0 p is a matrix norm.
Remark 2.15. The matrix p-norms are extremely useful because they satisfy the
following submultiplicative inequality.
Let A ∈ Cm×n and y ∈ Cn . Then
Ay p ≤A y p p. This is clearly true for y = 0, and for y = 0 it follows from
A p Ax p
yp = max
x =0 The matrix one norm is equal to the maximal absolute column sum.
Fact 2.16 (One Norm). Let A ∈ Cm×n . Then
m A 1 = max Aej
1≤j ≤n 1 = max 1 ≤j ≤n i =1 |aij |. Proof.
• The deﬁnition of p-norms implies
Hence A 1 1 = max
x Ax 1 =1 ≥ max1≤j ≤n Aej 1 ≥ Aej 1 ≤ j ≤ n. 1, 1. T • Let y = y1 . . . yn be a vector with A 1 = Ay 1 and y 1 = 1.
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