Proof of equality of OLS projection matrix and GLS projection matrix











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I'm struggling with the proof of the following proposition:



Given a $ntimes n$ symmetric, positive-semidefinite matrix $Omega$, a $ntimes k$ matrix $X$ such that $rank(X)=k$, and an invertible matrix $Q$ such that:
$$Omega X=XQ$$



Prove the following:
${{(X'{{Omega }^{-1}}X)}^{-1}}X'{{Omega }^{-1}}={{(X'X)}^{-1}}X'$



Here's my work before getting stuck:



enter image description here



I also tried playing with the Cholesky factorization of $Omega$ but always ended up getting stuck with an irreducible expression. What am I missing? (for context, this problem is from a graduate-level econometrics course)






linear-algebra matrices matrix-equations projection-matrices







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    up vote
    1
    down vote

    favorite
    1












    I'm struggling with the proof of the following proposition:



    Given a $ntimes n$ symmetric, positive-semidefinite matrix $Omega$, a $ntimes k$ matrix $X$ such that $rank(X)=k$, and an invertible matrix $Q$ such that:
    $$Omega X=XQ$$



    Prove the following:
    ${{(X'{{Omega }^{-1}}X)}^{-1}}X'{{Omega }^{-1}}={{(X'X)}^{-1}}X'$



    Here's my work before getting stuck:



    enter image description here



    I also tried playing with the Cholesky factorization of $Omega$ but always ended up getting stuck with an irreducible expression. What am I missing? (for context, this problem is from a graduate-level econometrics course)






    linear-algebra matrices matrix-equations projection-matrices







    share|cite|improve this question


























      up vote
      1
      down vote

      favorite
      1









      up vote
      1
      down vote

      favorite
      1






      1





      I'm struggling with the proof of the following proposition:



      Given a $ntimes n$ symmetric, positive-semidefinite matrix $Omega$, a $ntimes k$ matrix $X$ such that $rank(X)=k$, and an invertible matrix $Q$ such that:
      $$Omega X=XQ$$



      Prove the following:
      ${{(X'{{Omega }^{-1}}X)}^{-1}}X'{{Omega }^{-1}}={{(X'X)}^{-1}}X'$



      Here's my work before getting stuck:



      enter image description here



      I also tried playing with the Cholesky factorization of $Omega$ but always ended up getting stuck with an irreducible expression. What am I missing? (for context, this problem is from a graduate-level econometrics course)






      linear-algebra matrices matrix-equations projection-matrices







      share|cite|improve this question















      I'm struggling with the proof of the following proposition:



      Given a $ntimes n$ symmetric, positive-semidefinite matrix $Omega$, a $ntimes k$ matrix $X$ such that $rank(X)=k$, and an invertible matrix $Q$ such that:
      $$Omega X=XQ$$



      Prove the following:
      ${{(X'{{Omega }^{-1}}X)}^{-1}}X'{{Omega }^{-1}}={{(X'X)}^{-1}}X'$



      Here's my work before getting stuck:



      enter image description here



      I also tried playing with the Cholesky factorization of $Omega$ but always ended up getting stuck with an irreducible expression. What am I missing? (for context, this problem is from a graduate-level econometrics course)






      linear-algebra matrices matrix-equations projection-matrices




      linear-algebra matrices matrix-equations projection-matrices






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited yesterday

























      asked 2 days ago









      Maximiliano Santiago

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