Covariance matrix of 2 matrices?

In summary, the matrix equation tells us that the elements of \textbf{N} have unit variance and are uncorrelated. However, we cannot make any further deductions about the specific second order statistics of the elements.
  • #1
weetabixharry
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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!
 
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  • #2
weetabixharry said:
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
weetabixharry said:
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.
 

1. What is a covariance matrix?

A covariance matrix is a square matrix that summarizes the covariance of two or more variables. It is used to understand the relationship between variables and how they vary together.

2. How is a covariance matrix calculated?

A covariance matrix is calculated by taking the covariance between each pair of variables in a dataset and arranging them in a matrix format.

3. What does a positive covariance matrix indicate?

A positive covariance matrix indicates that the variables are positively correlated, meaning they tend to increase or decrease together.

4. Can a covariance matrix have negative values?

Yes, a covariance matrix can have negative values. This indicates that the variables are negatively correlated, meaning they tend to have an inverse relationship.

5. How is a covariance matrix used in data analysis?

A covariance matrix is used in data analysis to understand the relationship between variables and to determine which variables have the strongest correlation. It is also used in statistical modeling to make predictions based on the relationship between variables.

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