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Notation in linear algebra and rule for square of matrix norm

  1. Aug 7, 2011 #1
    Hi.

    I have a few simple questions.

    eq1.jpg (<- sorry, please click this image.)

    1. What does the notation in the red circle mean?

    2. Is there a rule for expanding square of norm? (e.g. || A*B*C ||^2)
    I don't really understand how the first eq. changes to the second eq.

    Thanks. :)
     
  2. jcsd
  3. Aug 7, 2011 #2

    micromass

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    Hi lishrimp! :smile:

    My guess:

    If [itex]\Sigma[/itex] is a matrix, then we can define a "norm" [itex]\|~\|_\Sigma[/itex] by setting

    [tex]\|\mathbf{x}\|_\Sigma=\sqrt{\mathbf{x}^T\Sigma \mathbf{x}}[/tex]

    In your case, the matrix is [itex]\Sigma^{-1}[/itex], so the norm is

    [tex]\|\mathbf{x}\|_{\Sigma^{-1}}=\sqrt{\mathbf{x}^T\Sigma^{-1} \mathbf{x}}[/tex]

    So

    [tex]\|\mathbf{x}\|_{\Sigma^{-1}}^2=\mathbf{x}^T\Sigma^{-1} \mathbf{x}[/tex]

    So that equality is true by definition.
     
  4. Aug 7, 2011 #3
    Thank you very much, micromass! :D
     
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