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Math Help - Subordinate Matrix Norm equivalences.

  1. #1
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    Subordinate Matrix Norm equivalences.

    Prove that for any matrix A and any vector norm ||·|| the following definitions of subordinate matrix norm ||A|| are equivalent...

    ||A|| = max(x!=0) ||Ax||/||x||

    and

    ||A|| = max(||u||=1) ||Au||

    I know I probably have to use the definitions of a matrix norm but I'm stumped. I don't really understand the topic all that well and none of the information I've found is very helpful.

    Once again, any help greatly appreciated and sorry about the rubbish formatting.
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  2. #2
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    \sup_{V- \{ 0 \} } \frac{ \Vert Ax \Vert}{ \Vert x \Vert } = \sup_{V- \{ 0 \} } \Vert A( \frac{x}{ \Vert x \Vert}) \Vert  = \sup_{S_V} \Vert Ax \Vert where S_V is the unitary sphere of V. The first equality follows from the properties of the norm and the fact that A is linear.
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