Relation between Gram matrix distributions

[nikozm]

Hello,

Assume that H is a n \times m matrix with i.i.d. complex Gaussian entries each with zero mean and variance \sigma. Also, let n>=m. I ' m interested in finding the relation between the distribution of H^HH and HH^H, where ^H stands for the Hermittian transposition. I anticipate that both follow the complex wishart distribution with the same parameters (since they share the same nonzero eigenvalues), but I m not sure about this.

Any ideas ? Thanks in advance..

 

[mmwave]

Hi there,

Thank you for your interesting question. You are correct in your intuition that both HHH and HHH follow the complex Wishart distribution with the same parameters. This can be proven using the properties of the Hermittian transposition and the complex Wishart distribution.

First, let's define some notation. Let HH be a complex Gaussian matrix with zero mean and variance \sigma, as described in the forum post. We can write HH as HH = XX^H, where X is a complex Gaussian matrix with zero mean and variance \sigma. The matrix X^H is the conjugate transpose of X. We can also define HHH as HHH = YY^H, where Y is the Hermittian transposition of X.

Now, the complex Wishart distribution is defined as the distribution of HH = XX^H, where X is a p \times n matrix with i.i.d. complex Gaussian entries each with zero mean and variance \sigma. This means that both HH and HHH follow the same distribution, since they are both of the form XX^H.

Furthermore, the nonzero eigenvalues of a complex Wishart matrix are distributed according to the chi-square distribution with 2n degrees of freedom, where n is the number of rows of the matrix. Since both HH and HHH have the same nonzero eigenvalues, they must also follow the same chi-square distribution with 2n degrees of freedom.

In conclusion, both HH and HHH follow the complex Wishart distribution with the same parameters \sigma and 2n degrees of freedom. I hope this helps answer your question. Let me know if you have any further doubts or if you need any clarification.