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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 p k =  k p k for all 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 = and thus k k = k k =  k k for any R , from which it follows that k k = 0 if we have chosen 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. 238239 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 1norm: k x k 1 = n i =1  x i  . Euclidean norm or 2norm: k x k 2 = x T x = p n i =1 x 2 i pnorm: 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. 403404 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. 405406 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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 Summer '08
 KIM
 Linear Algebra, Algebra

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