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Proving a linear algebra equation

  1. Jul 1, 2015 #1
    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 + Σε


    B = (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^-1


    Assumptions

    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
     

    Attached Files:

  2. jcsd
  3. Jul 1, 2015 #2

    HallsofIvy

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    "Sigma", [itex]\Sigma[/itex], here, is a matrix, not a summation symbol?
     
  4. Jul 1, 2015 #3
    Yes, HallsofIvy, you are correct...
     
  5. Jul 2, 2015 #4
    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.
     
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