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A question concerning matrix norms

  1. Oct 14, 2007 #1
    I wonder given [tex]||Ax||_{a} = ||Bx||_{b}[/tex] for any [tex]x \in \mathbb{R}^n[/tex], is it true that [tex]||A||_{a} = ||B||_{b}[/tex], where [tex]||.||_{a}, ||.||_{b}[/tex] are two vector norms and the matrix norms are induced by the corresponding vector norms?

    Thanks in advance.
     
  2. jcsd
  3. Oct 14, 2007 #2

    morphism

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    Yes. Just write down the definition of ||A||_a and show that it's equal to ||B||_b.
     
  4. Oct 17, 2007 #3
    The matrix norm induced by a vector norm [tex]||.||_a[/tex] is defined by
    [tex]||A||_a = \sup_{x \neq 0} \frac{||Ax||_a}{||x||_a}[/tex]. In showing the equality of matrix norms under the condition I've posted last time, I don't know how to deal with the denominator appeared in the denominator inside the supremum as [tex]||x||_a[/tex] and [tex]||x||_b[/tex] may be of different values.
     
    Last edited: Oct 17, 2007
  5. Oct 17, 2007 #4

    radou

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    Are you referring to operator norms here?
     
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