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Ans 4 - v1 - Matrix Norms...

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ne previo Next: Direct Methods of Solving Up: Norms and Convergence Previous: Norms of Vectors Because we deal with equations like which involve a matrix as well as vectors, we extend norms to matrices. We could of course regard as a vector (with some order specified) in , but since matrix multiplication may arise, we add an extra property. Definition 2.2.1 A matrix norm on is a mapping with the properties: (i) , (ii) , (iii) , (iv) . Not all vector norms will become matrix norms in because of axiom (iv). Most of the norms we use here are derived from vector norms in the following way: Definition 2.2.2 The matrix norm induced by or subordinate to the given vector norm is given by (2.1) We can regard as the maximum `magnification' capability of A . Note that replacing by in this definition makes no difference; if we choose so that , we obtain the alternative definition Matrix Norms http://www.maths.lancs.ac.uk/~gilbert/m306a/node6.html 1 of 10 10/22/10 11:43 PM
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This sup is attained if is a continuous function of the components of , since the surface of the unit ball is compact. Define by ; then we show that is a vector semi-norm. (i) . Unfortunately does not imply that , so is not a norm, but a semi-norm. However a semi-norm possesses the bound attainment property, so we can work with this. (ii) . (iii) . Thus is a semi-norm and hence attains its maximum on the unit ball. We may now write the last definition in the form (2.2) Finally, when a lot of manipulation is to be performed, we can write the last definition as where is the value of in the previous definition at which the maximum is attained. Clearly .
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