# Variance of a vector product

1. Dec 11, 2014

### nikozm

Hi,

Assuming that A is a n x m random matrix and each of its entries are complex Gaussian with zero mean and unit-variance. Also, assume that b is a n x1 random vector and its entries are complex Gaussian with zero mean and variance=s. Then, what would be the variance of their product Ab?

Any help would be useful.

Thanks

2. Dec 16, 2014

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Dec 19, 2014

### coquelicot

I think that you have first to know the variance of a product of two gaussians. More precisely, if X and Y are two independant random variables with distribution N(0,s) and N(0,s'), then what is the distribution of XY ? There are formulae in the litterature (Google it). Once you have obtained the distribution Z of XY, you have to know the distribution of the sum of several variable Z' of the same type (theoretically, this is the convolution product of the variables). I am almost certain that Z, and the sum of the variables Z', will have zero mean. The variance should be given in the litterature, if the sum of the Z' is a known distribution. If your matrix is large, then you can use the central limit theorem to approximate the sum of the Z'.

Last edited: Dec 19, 2014