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Covariance matrix of 2 matrices?

  1. Nov 10, 2011 #1
    I have an [itex](m \times n)[/itex] complex matrix, [itex]\textbf{N}[/itex], whose elements are zero-mean random variables. I have a sort of covariance expression:

    [itex]\mathcal{E}\left\{\textbf{N}\textbf{N}^H\right\} = \textbf{I}[/itex]

    where [itex]\mathcal{E}\left\{\right\}[/itex] denotes expectation, [itex]\{\}^H[/itex] is conjugate transpose and [itex]\textbf{I}[/itex] is the identity matrix.

    Basically, I want to know exactly what this tells me about the second order statistics of the elements of [itex]\textbf{N}[/itex]. For example, I know that if instead I just had an [itex](m \times 1)[/itex] vector, [itex]\textbf{n}[/itex], then an identity covariance matrix would imply that all the elements of [itex]\textbf{n}[/itex] have unit variance and are uncorrelated.

    Can I make any similar deductions from the matrix equation, above? Many thanks for any help!
  2. jcsd
  3. Nov 10, 2011 #2


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    Hey weetabixharry.

    To me an identity covariance matrix says that there is no covariance terms at all for the different permutations, but I would want to see the expanded definitions for each random variable just to be absolutely sure.

    I'm not familiar with your matrix identity though, so I would need to put that into context of standard results concerning expanded definitions of the associated random variables.
  4. Nov 11, 2011 #3

    D H

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    The covariance of an NxN matrix is a NxNxNxN cartesian tensor; a very ugly beast. Your NxN matrix is the equivalent of the diagonal of a covariance matrix for a Nx1 vector.
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