# A question on matrix calculation

## Recommended Posts

ronm    105
Hello all, let say I have a PD symmetric matrix Sigma (order n) and two matrices A and B as order rXn. Now consider the expression:

(A * Sigma * A') - (B * Sigma * B')

Can I write this expression as (C * Sigma * C')?

##### Share on other sites
alvaro    21246
No, you can't in general do that. If n < r, (A * Sigma * A') - (B * Sigma * B') will generally have rank 2*n, but (C * Sigma * C') can only have rank n.

##### Share on other sites
ronm    105
okay, let say r = 1 and n > 1. In this restricted scenario what will be the case?

Thanks,

##### Share on other sites
Brother Bob    10344
If by PD you mean positive definite (I searched, but no other relevant terms showed up, just disregard if you mean something else), then no, you cannot find such a relation. You have two quadratic forms with A and B that are always positive since Sigma is positive definite. The difference between them may be negative if the quadratic form of B is greater than the one of A. Since sigma is positive definite, you cannot find a C that results in a negative value.

##### Share on other sites
alvaro    21246
If r=1, (A * Sigma * A') - (B * Sigma * B') is a 1x1 matrix. If its element happens to be positive, then yes, you can find a vector C (many, actually) such that (C * Sigma * C') has the same value.

How about you tell us what you are trying to do, instead of going back and forth with descriptions of the problem that don't quite make sense?

##### Share on other sites
ronm    105
Thanks for your comment. Basically I want to have a Statistical test as H0: A * Sigma * A' = B * Sigma * B'

Here the only population parameter is Sigma, which can be estimated from sample observations. Therefore I know the distribution of Sigma_Sample. And from this I can derive the distribution of C * Sigma * C'.

Therefore if can somehow write (A * Sigma * A') - (B * Sigma * B') = C * Sigma * C' then I can construct a Test Statistic for my testing.

It will also be very helpful if you can propose some better alternative of my testing problem.

Thanks and regards,

##### Share on other sites
ronm    105
any suggestion pls?

Thanks,

##### Share on other sites
Inferiarum    739
Why can you not derive the distribution of (A * Sigma * A') - (B * Sigma * B')?