Relation between covariance and rank

In summary, covariance is a measure of the relationship between two variables and is calculated by taking the average of the products of the deviations of the values from their respective means. It is related to correlation, which is a standardized version of covariance. The formula for calculating covariance involves the individual data points, means, and number of data points. The rank of variables does not affect covariance, but it can affect correlation. A covariance of 0 indicates no linear relationship between two variables, but this does not necessarily mean there is no relationship at all between the variables.
  • #1
nikozm
54
0
Hi,

Assume a matrix H n\times m, with random complex Gaussian coefficients with zero-mean and unit-variance. The covariance of this matrix (i.e., expectation [HHH]) assuming that m = 1 is lower than another H matrix when m > 1 ??

If this holds, can anyone provide a related reference?

Thanks in advance
 
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  • #2
What does HHH mean? HHT?

How do you compare two covariance matrices to say which one is "lower"?
 

1. What is the definition of covariance?

Covariance is a measure of how two variables change together. It measures the direction and strength of the relationship between two variables.

2. How is covariance related to correlation?

Covariance and correlation both measure the relationship between two variables, but correlation is a standardized version of covariance that falls between -1 and 1, making it easier to interpret. A covariance of 0 indicates no linear relationship, while a correlation of 0 indicates no linear relationship and no relationship of any kind.

3. What is the formula for calculating covariance?

The formula for calculating covariance is: Cov(X,Y) = Σ (xi - x̄)(yi - ȳ) / N, where xi and yi are the individual data points, x̄ and ȳ are the means of X and Y, and N is the number of data points.

4. How does the rank of variables affect the covariance?

The rank of variables does not affect the covariance. Covariance is a measure of the relationship between the actual values of two variables, not their ranks. However, rank can affect the correlation between two variables, as correlation is based on the standardized values of the variables.

5. What does a covariance of 0 indicate?

A covariance of 0 indicates no linear relationship between two variables. However, it is important to note that this does not necessarily mean there is no relationship at all between the variables, as they may have a nonlinear relationship or a relationship that is not captured by covariance.

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