Covariance matrix of 2 matrices?

  • #1

Main Question or Discussion Point

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!
 

Answers and Replies

  • #2
chiro
Science Advisor
4,790
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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!
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.
 
  • #3
D H
Staff Emeritus
Science Advisor
Insights Author
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Can I make any similar deductions from the matrix equation, above? Many thanks for any help!
No.

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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