Fact 213 if a e cmn then ae a e proof the triangle

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Unformatted text preview: ˜ 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 ˜ −1 ˜ 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 143.215.200.123. 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 Ay p ≥ . xp 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 A 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. Viewing the matrix vector product...
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