# Matrix notation - Two jointly Gaussian vectors pdf

1. Aug 1, 2011

### 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

File size:
52 KB
Views:
137
2. Aug 1, 2011

### 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)

3. Aug 2, 2011

### 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??

File size:
23.5 KB
Views:
79
4. Aug 2, 2011

### 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].

5. Aug 3, 2011

### EmmaSaunders1

I have managed to obtain the desired result - it was simply grouping the matrix multiplication into two parts seperated 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???