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 HHH and HHH, 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..

 

[jvicens]

Hello,

Thank you for your question. The relation between the distributions of HHH and HHH is indeed complex wishart distribution with the same parameters. This is because both HHH and HHH have the same nonzero eigenvalues, which is a property of the complex wishart distribution. The complex wishart distribution is a generalization of the real wishart distribution and is commonly used in statistical signal processing and multivariate analysis.

To understand this further, let's first define the complex wishart distribution. It is a probability distribution over complex matrices with positive definite Hermitian matrices as its support. It is also known as the complex Gaussian distribution of the second kind. The complex wishart distribution is characterized by two parameters, n and \sigma, where n is the degree of freedom and \sigma is the covariance matrix. In your case, n is equal to the number of rows in H and \sigma is the variance of the complex Gaussian entries.

Now, let's look at the relation between HHH and HHH. As mentioned earlier, both matrices have the same nonzero eigenvalues. This means that the eigenvalues of HHH and HHH are equal, and therefore, they have the same distribution. Since the distribution of HHH is complex wishart with parameters n and \sigma, the distribution of HHH will also be complex wishart with the same parameters.

I hope this helps to clarify your doubts. If you have any further questions, please feel free to ask. Thank you.