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Math Help - Operator norm

  1. #1
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    Oct 2008
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    Operator norm

    I'm a bit confused about what an operator norm is.

    Can someone explain it to me using a simple matrix?

    What value would you use for the Euclidean norm (x) when it is multiplied by A? (before taking the supremum).
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  2. #2
    Junior Member
    Joined
    Mar 2010
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    I'm not sure which definition you're using for the operator norm, but the definition I use is if (X, \|\cdot\|_X), (Y, \|\cdot\|_Y) are normed vector spaces and T : X \to Y is linear and bounded, define \|T\|_{B(X,Y)} := \inf(\{M \in [0,\infty) : \forall x \in X, \|T(x)\|_Y \leq M \|x\|_X\}. I assume you're using the formula \|T\|_{B(X,Y)} = \sup(\{ \|T(x)\|_Y : x \in X, \|x\| \leq 1\}).

    Define A := \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} and let (x,y)^t \in \mathbb{R}^2 be given such that \|(x,y)^t\| \leq 1. Then \|A (x,y)^t\| = \|(y, -x)^t\| = \|(x,y)^t\|. Therefore \|A\|_{B(X,Y)} = \sup(\{\|(x,y)^t\| : (x,y)^t \in \mathbb{R}^2, \|(x,y)^t\| = 1\}) = 1.
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