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IlyaMath
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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
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
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