571_7.4

# 571_7.4 - 7.4 MATRIX NORMS AND CONDITION NUMBERS KIAM HEONG...

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Unformatted text preview: 7.4 MATRIX NORMS AND CONDITION NUMBERS KIAM HEONG KWA p. 237, p. 404 A linear space V is called a normed linear space if there is a func- Normed linear space tion k · k : V → R , called a norm on V , satisfying Nonnegativity: k p k ≥ for all p ∈ V ; Positivity: k p k = 0 only if p = ; Homogeneity: k α p k = | α |k p k for all α ∈ R and all p ∈ V ; Triangle inequality: k p + q k ≤ k p k + k q k for all p , q ∈ V . As a remark, the homogeneity implies that k k = 0 . To see this, recall that α = and thus k k = k α k = | α |k k for any α ∈ R , from which it follows that k k = 0 if we have chosen α 6 = 0 . Also, for any p , q ∈ V , it is customary to call k p- q k as the distance between p and q . pp. 238-239 Let x = ( x 1 ,x 2 , ··· ,x n ) T ∈ R n .Each of the following is a norm Norms on R n on R n : Taxicab norm or Manhattan norm or 1-norm: k x k 1 = ∑ n i =1 | x i | . Euclidean norm or 2-norm: k x k 2 = √ x T x = p ∑ n i =1 x 2 i p-norm: For any p ∈ R , p ≥ 1 , k x k p = ( ∑ n i =1 | x i | p ) 1 /p . Uniform norm or infinity norm: k x k ∞ = max 1 ≤ i ≤ n | x i | . pp. 403-404 Let A = ( a ij ) ∈ R m × n , i.e., let A be an m × n matrix. The norm on Frobenius norm R m × n given by k A k F = n X j =1 m X i =1 a 2 ij ! 1 / 2 = m X i =1 n X j =1 a 2 ij ! 1 / 2 is called the Fronenius norm . The following properties of k · k F can be established easily: (1) k A k F = ∑ n j =1 k a j k 2 2 1 / 2 , where a j is the j th column of A . (2) k A k F = ∑ m i =1 k ~ a T i k 2 2 1 / 2 , where ~ a i is the i th row of A . (3) For any x ∈ R n , k A x k 2 ≤ k A k F k x k 2 . (4) For any B ∈ R n × r , k AB k F ≤ k A k F k B k F . Date : July 10, 2011. 1 2 KIAM HEONG KWA Property (3) is the compatibility condition on the norms k · k F on R m × n and k · k 2 on R n , while property (4) is called submultiplica- tivity . The submultiplicativity is a desired and pleasant property on a matrix norm because, for instance, in the case of the Frobenius norm on R n × n , we have k A k k F ≤ k A k k F for any A ∈ R n × n and for all integer k ≥ 1 , so that if k A k F < 1 , then lim k →∞ k A k k F = 0 . pp. 405-406 Generally, a matrix norm k · k M on R m × n and a vector norm k · k V on R n are said to be compatible provided Compatible norms on R m × n and R n k A x k V ≤ k A k M k x k V for any A ∈ R m × n and any x ∈ R n . Convention: We will drop the subscript in the norm notation in the sequel, relying on the context to render it clear the norm under consideration As a matter of fact, for each vector norm k · k on R n , there is Theorem 7.4.1 always one matrix norm k · k on R m × n such that the norms are compatible. Such a matrix norm is given by k A k = max x ∈ R n x 6 = k A x k k x k for all A ∈ R m × n . Convince yourself that this is really a norm on R m × n . The matrix norm obtained this way is said to be subordinate to the vector norm. It is clear from the definition thatto the vector norm....
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571_7.4 - 7.4 MATRIX NORMS AND CONDITION NUMBERS KIAM HEONG...

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