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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)  αv  =  α  v  for α ∈ 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 supnorm: n X i =1 a i e i sup = max i  a i  . This gives rise to the supmetric 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 supnorm 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) positivedefinite: ( 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 CauchySchwarz 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 CauchySchwarz inequality is true for any inner product on a realvector 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 supnorm on R n does not arise from an inner product, i.e. there is no inner product whose associated norm is the supnorm. Although the supnorm 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
 Chandrasekara
 Linear Algebra, Vector Space, Norm, Hilbert space, inner product

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