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Homework Help: How can this be equal to the unit matrix?

  1. Jan 21, 2010 #1
    1. The problem statement, all variables and given/known data
    At the lecture yesterday the teacher just ended up with a result I could not arrive at. So, how
    can the below stated expression be verified?

    [tex]\left(C^{1/2}\right)^{T}C^{-1}C^{1/2}=I[/tex]
    Here C is a nonsingular covariance matrix, obviously, and I is the unit matrix.

    I will not make an attempt of a solution because then it feels like I would solve it but not understand. I hope that is ok. What I seek here is not an rigorous proof. I just want to understand.

    Hope someone can help me!
     
  2. jcsd
  3. Jan 21, 2010 #2

    D H

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    Suppose some invertible matrix n×n C is decomposed into the product of two n×n matrices A and B:

    [tex]C = AB[/tex]

    Then A and B must themselves be invertible and the inverse of C is given by

    [tex]C^{-1} = B^{-1}A^{-1}[/tex]

    The definition of the matrix square root of some matrix C is that

    [tex]C=\left(C^{1/2}\right)^T\,C^{1/2}[/tex]

    Combine the above two and the result in the original post falls right out.
     
  4. Jan 21, 2010 #3
    If I understand you correctly, then it is allowed to change the order in the matrix multiplication?
    [tex]C^{-1}C^{1/2}=C^{1/2}C^{-1}[/tex]
     
  5. Jan 21, 2010 #4
    There was no need to change the order. Now I see. :) Thanks!
     
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