# Covariance matrix of 2 matrices?

1. Nov 10, 2011

### weetabixharry

I have an $(m \times n)$ complex matrix, $\textbf{N}$, whose elements are zero-mean random variables. I have a sort of covariance expression:

$\mathcal{E}\left\{\textbf{N}\textbf{N}^H\right\} = \textbf{I}$

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

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

Can I make any similar deductions from the matrix equation, above? Many thanks for any help!

2. Nov 10, 2011

### chiro

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. Nov 11, 2011

### D H

Staff Emeritus
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.