Proving a linear algebra equation

[IlyaMath]

I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent.

I would appreciate any help on this.

Prove A = B

A = (Σπ^-1 + Σy^-1)^-1 * (Σπ^-1*π + Σy^-1*y)

y = π+ X*β

Σπ =τ*Σ

Σy = X' * Σβ * X + ΣεB = (Σπ^-1 + P'*Σβ^-1*P)^-1 * (Σπ^-1*π + P'*Σβ^-1*q)

q = P*y

P = (X'*Σ^-1*X)^-1*X'*Σ^-1Assumptions

i) Σε is infinitesimally small. (If Σε is exactly zero, then Σy may not be invertible).

ii) N > F (If N = F, then the proof is trivial. If N < F then is probably not invertible and P is not defined.)Notation

A, B: Nx1

π: Nx1

y: Nx1

q: Fx1

β: Fx1

Σ: NxN

Σπ: NxN

Σy: NxN

Σβ: FxF

Σε: NxN

X: NxF

P: FxN

τ: 1x1

 

[HallsofIvy]

"Sigma", $\Sigma$, here, is a matrix, not a summation symbol?

 

[IlyaMath]

Yes, HallsofIvy, you are correct...

 

[IlyaMath]

This problem comes from Bayesian statistics, where N is number of observations and F is number of factors. All of the Sigma (Σ) are variance-covariance matrices. The Greek characters after each Σ are meant to be subscripts.