Matrix notation - Two jointly Gaussian vectors pdf

[EmmaSaunders1]

Hello

I am having trouble deriving using block matrix notation the conditional pdf of two joint Gaussian vectors:

I assume that it just involves some re-arranging of eq 1 (attatched) but am unsure if taking the inverse of the resultant matrix in eq 1 is valid and if the order of multiplication holds.

Thoughts appreciated

 

[bpet]

There's an error, inv([I,0;A,I])=[I,0;-A,I], otherwise looks on the right track. If you expand the product it should get an answer that is symmetric in x and y as the factorization shouldn't matter. The Woodbury matrix identity may or may not be useful here.

And yes the matrix isn't guaranteed to be invertible (e.g. if X=Y)

 

[EmmaSaunders1]

Hello

Thanks very much for your help. I have multiplied out the problem and looked for symmetry as you suggested. I do however have an extra term in comparison to the final solution;

Would you possibly be-able to take a look at the attatched - perhaps I am missing something - is there any kind of concept or theorem I am missing which suggests the extra term is zero or is fundamentally the calculation wrong. I notice in the original attachment there is a "X" sign I assumed this to be matrix multiplication rather than cross product - is this correct??

Thanks again for your help

 

[bpet]

Sorry the original version looks correct, ignore my previous comments - in effect you're showing that [I,-Sxx*inv(Syy);0,I]*[x;y] given y is gaussian. Notice that the exponent reduces to -(1/2)*[x'-xbar',y'-ybar']*(inv(S)-[0,0;0,inv(Syy)])*[x-xbar;y-ybar] and use [0,0;0,inv(Syy)] = [I,0;-inv(Syy)*Syx,I]*[0,0;0,inv(Syy)]*[I,-Sxx*inv(Syy);0,I].

 

[EmmaSaunders1]

Hi Thanks for your help:

I have managed to obtain the desired result - it was simply grouping the matrix multiplication into two parts separated by the X sign in the first attachment to make the multiplication easier. Would you however please be able to clarify - during the expansion I assumed that the product of two different covariance matrices are commutive - is this assumption okay.

I would also like to understand the simpler way you have tried to explain but am unable to follow the logic of the substitution as shown on the attached?

Your helps appreciated

Thanks