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Matrix Identify involving Diagonal Matrix

  1. Nov 15, 2008 #1

    hotvette

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    If V is diagonal, it is easy to show:

    [itex](V + V^{-1})^{-1} = V(V^2 + I)^{-1}[/itex]

    by multiplying both sides by:

    [itex](V + V^{-1})[/itex]

    But, I'm wondering if there is a way to derive the RHS from the LHS. Since diagonal matrices behave like scalars, I used a scalar analogy:

    [itex] (x + 1/x)^{-1} = ((x^2 + 1)/x)^{-1} = x(x^2 + 1)^{-1}[/itex]

    But I'd like to show it using matrix terminology. If there is something simple, I'm missing it. Any ideas?
     
  2. jcsd
  3. Nov 15, 2008 #2

    morphism

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    V doesn't even have to be diagonal, only invertible (and we also need V^2+I to be invertible - if we're working with real matrices, this is automatic).

    [tex](V + V^{-1})^{-1} = ((V^2 + I)V^{-1})^{-1} = V(V^2 + I)^{-1},[/tex]

    where in the middle step I'm pulling out the "common factor" [itex]V^{-1}[/itex], and in the last step I'm using the fact that [itex](AB)^{-1} = B^{-1}A^{-1}[/itex].
     
    Last edited: Nov 15, 2008
  4. Nov 15, 2008 #3

    hotvette

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    I knew it had to be simple. Just didn't see it....

    Thanks.
     
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