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Thread: 2-norm of a matrix

  1. #1
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    2-norm of a matrix

    How do you prove that

    $\displaystyle \left\|A^{*} A \right\|_{2}= \left\| A \right\|^{2}_{2}$ ?

    I can prove that $\displaystyle \left\|A^{*} A \right\|_{2} \leq \left\| A \right\|^{2}_{2}$

    but I am not sure how to proceed for the other inequality.
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  2. #2
    Junior Member nimon's Avatar
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    Which norm are you using for these matrices? Is it

    $\displaystyle
    \left\| A \right\|_{2} = \left( \sum\limits_{i,j} |A_{ij}|^{2} \right)^{\frac{1}{2}}
    $

    ?
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  3. #3
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    2-norm of a matrix

    I think

    $\displaystyle \left\| A \right\| _{2} = Sup_{x \neq 0} \frac {\left\| Ax \right\| _{2}}{\left\| x \right\| _{2}} $

    and the one that you had might be the Frobenius norm.
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