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Unformatted text preview: COMPUTING THE NORM OF A MATRIX KEITH CONRAD 1. Norms on Vector Spaces Let V be a vector space over R . A norm on V is a function |||| : V R satisfying three properties: 1) || v || 0, with equality if and only if v = 0, 2) || v + w || || v || + || w || for v,w V , 3) || v || = | ||| v || for R , v V . The same definition applies to a complex vector space. From a norm we get a metric on V by d ( v,w ) = || v- w || . The standard norm on R n is n X i =1 a i e i = v u u t n X i =1 a 2 i . This gives rise to the Euclidean metric on R n . Another norm on R n is the sup-norm: n X i =1 a i e i sup = max i | a i | . This gives rise to the sup-metric on R n : d ( a i e i , b i e i ) = max | a i- b i | . On C n the standard norm is n X i =1 a i e i = v u u t n X i =1 | a i | 2 , and the sup-norm is defined as on R n . A common way of placing a norm on a real vector space V is via an inner product , which is a pairing ( , ): V V R that is 1) bilinear, 2) symmetric: ( v,w ) = ( w,v ), and 3) positive-definite: ( v,v ) 0, with equality if and only if v = 0. The standard inner product on R n is n X i =1 a i e i , n X i =1 b i e i ! = n X i =1 a i b i . For an inner product ( , ) on V , a norm can be defined by the formula || v || = p ( v,v ) . That this actually is a norm is a consequence of the Cauchy-Schwarz inequality, | ( v,w ) | p ( v,v )( w,w ) = || v |||| w || , whose proof can be found in most linear algebra books. In 1 2 KEITH CONRAD particular, using the standard inner product on R n we get the classic form of this inequality, as proven by Cauchy: n X i =1 a i b i v u u t n X i =1 a 2 i n X i =1 b 2 i . However, the Cauchy-Schwarz inequality is true for any inner product on a real-vector space, not just the standard inner product on R n . The norm on R n that comes from the standard inner product is the standard norm. On the other hand, the sup-norm on R n does not arise from an inner product, i.e. there is no inner product whose associated norm is the sup-norm. Although the sup-norm and the standard norm on R n are not equal, they are each bounded by a constant multiple of the other one: max i | a i | v u u t n X i =1 a 2 i n max i | a i | , i.e. || v || sup || v || n || v || sup . Therefore the metrics these two norms give rise to determine the same notions of convergence: a sequence in R n which is convergent with respect to one of the metrics is also convergent with respect to the other metric. The standard inner product on R n is closely tied to transposition of n by n matrices. For A = ( a ij ) M n ( R ), let A > = ( a ji ) be its transpose. Then for any v,w R n , ( Av,w ) = ( v,A > w ) , where ( , ) is the standard inner product....
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- Fall '08