# Distribution of 2 matrices with the same eigenvalues

1. Nov 24, 2015

### nikozm

Hi,

I was wondering if two matrices with the same eigenvalues share the same PDF.

Any ideas and/or references would be helpful.

2. Nov 24, 2015

### andrewkirk

Matrices in general don't have anything that is widely referred to as a 'PDF'.
The only PDF I know is 'probability density function' in probability theory. Is that what you mean? If so, how do you want to relate it to a matrix? Matrices are used in probability theory and statistics in numerous different ways.

3. Nov 25, 2015

### 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..

4. Nov 25, 2015

### andrewkirk

The matrix $H^HH$ will be $m\times m$ while $HH^H$ will be $n\times n$. They will have different numbers of eigenvalues. Why do you think the nonzero ones will be the same?

5. Nov 25, 2015

### nikozm

Indeed, they have different dimensions. However their non-zero eigenvalues are the same. This is a fact. If you hold reservations about the latter just implement it in Matlab and see with the command eig their corresponding eigenvalues.

My question is: if they also have the same probability density function.