How to prove a non negative definite matrix has to be a variance-covariance matrix?

[arsene_lady]

I know this is a iff condition,and the reverse is easy to prove.But I am having some difficulty to prove that all non negative definite matrices are variance covariance matrix.From what i have collected from wikipedia,I tried to prove it this way.

Let A be a non negative definite matrix.
Then A can be written as

 A=BB^t,B^t is the tranpose of B.
ie A=BIB
    =Var(BX),where X is a random variable whose variance-covariance matrix is identity matrix I.

But this method is not generalising the proof for any variance covariance matrix.So is this approach correct? Also is there any better way to solve this?

 

[BGM]

Re: How to prove a non negative definite matrix has to be a variance-covariance matri

You have already answer the question, as you have find out the corresponding variance-covariance matrix of random vector for each non-negative definite matrix.

Of course you may also want to ask "Are all variance-covariance matrix non-negative definite?".I am sure you can answer this question easily as well.

 

[arsene_lady]

Re: How to prove a non negative definite matrix has to be a variance-covariance matri

The approach of proof I tried, only proves that a non negative definite matrix is a variance covariance matrix of a set of linear combinations of some random variable whose variance covariance matrix is the identity matrix.So is this not becoming a particular case?

 

[BGM]

Re: How to prove a non negative definite matrix has to be a variance-covariance matri

You may restate your question as

Given a non-negative definite matrix, is there exist some random vectors such that its variance-covariance matrix equals to the given matrix?

So by construction you have find it out and it always exist.

 

[arsene_lady]

Re: How to prove a non negative definite matrix has to be a variance-covariance matri

Thanks,confusion cleared. I have undersood what you are trying to say.