# Matrix Integral of different size

1. Nov 18, 2012

### quantumlight

I have this problem:

1. θ is [θ0, θ1]T and θreplicated = [θ0, θ0, θ1, θ1]T, further θ is a gaussian with mean mθ and covariance matrix S.

2. V is [V1, V2, V3, V4]T

3. N is [N1, N2, N3, N4]T, further it has gaussian distribution with mean 0, and diagonal covariance matrix σ.

4. let r = diag(V)*(1+jθreplicated) + N where p(r|v, θ) is a gaussian with mean diag(V)*(1+jθreplicated) and diagonal covariance matrix σ

If I know r, how do I integrate p(θ)[(r-diag(V) - jdiag(V)θreplicated]H[(r-diag(V) - jdiag(V)θreplicated]dθ?

I know p(θ)[(r-diag(V) - jdiag(V)θ]H[(r-diag(V) - jdiag(V)θ]dθ -> if θ is not replicated would be tr(ASA) + (Amθ+b)H(Amθ+b) where A is -jdiag(V), and b is r - diag(V), but how would this work with θreplicated?