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Math Help - Norm in Matrix algebra

  1. #1
    Member Mauritzvdworm's Avatar
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    Norm in Matrix algebra

    let A\in M_n(\mathbb{C}), show that
    \|A\|^2=\max\{\lambda: \det(\lambda-A^*A)=0\}=\max\{\lambda: \det(\lambda-AA^*)=0\}
    Last edited by Mauritzvdworm; July 14th 2010 at 01:38 AM. Reason: tipo
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    let A\in M_n(\mathbb{C}), show that
    \|A\|^2=\max\{\lambda: \det(\lambda-A^*A)=0\}=\max\{\lambda: \det(\lambda-AA^*)=0\}
    The proof consists of two steps. (1) \|A\|^2 = \|A^*A\|; (2) A*A is a positive definite matrix and so its norm is equal to its largest eigenvalue. Those two steps together show that \|A\|^2=\max\{\lambda: \det(\lambda-A^*A)=0\}.

    For the last part, use the fact that \|A\| = \|A^*\| to deduce that \|A\|^2=\max\{\lambda: \det(\lambda-AA^*)=0\}.
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