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:
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
asked Nov 19 at 1:02
Maximiliano Santiago
1077
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up vote
1
down vote
favorite
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:
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
asked Nov 19 at 1:02
Maximiliano Santiago
1077
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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:
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
asked Nov 19 at 1:02
Maximiliano Santiago
1077
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:
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
asked Nov 19 at 1:02
Maximiliano Santiago
1077
asked Nov 19 at 1:02
Maximiliano Santiago
1077
edited Nov 19 at 17:45
asked Nov 19 at 1:02
Maximiliano Santiago
1077
asked Nov 19 at 1:02
Maximiliano Santiago
1077
asked Nov 19 at 1:02
Maximiliano Santiago
1077
1077
add a comment |
add a comment |
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