Relation between covariance and rank

[nikozm]

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 [HH^H]) 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

 

[mfb]

What does HH^H mean? HH^T?

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