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ronm

A question on matrix calculation

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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')?

Thanks for your help.

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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.

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okay, let say r = 1 and n > 1. In this restricted scenario what will be the case?

Thanks,

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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.

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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?

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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.

Please let me know if you need more information.

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

Thanks and regards,

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