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 2 years ago
Linear Algebra/Linear Regression:
The Hat matrix is defined as \(H=X(X^TX)^{1}X^T\). Messing with it a bit I found that it was equal to the identity matrix o_O? Can you show me where I went wrong?
 2 years ago
Linear Algebra/Linear Regression: The Hat matrix is defined as \(H=X(X^TX)^{1}X^T\). Messing with it a bit I found that it was equal to the identity matrix o_O? Can you show me where I went wrong?

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kirbykirby
 2 years ago
Best ResponseYou've already chosen the best response.1It shouldn't be though since this matrix is significant for linear regression. (I mean why would they define such a long expression?). \[X(X^TX)^{1}X^T\]=\(X(X^{1}(X^T)^{1})X^T\)\[=X(X^{1}(X^{1})^{T})X^T\]\[=XX^{1}(X^{1})^TX^T\]\[=I(X^{1})^TX^T\]\[=I(XX^{1})^T\]\[=II=I\]

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.1Took a little research, but you're correct. However, if \(X\) does not have an inverse, this does not result in the identity.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.1See here for a little more information http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse

kirbykirby
 2 years ago
Best ResponseYou've already chosen the best response.1o_o interesting. Hmm lol this Hat matrix is a lot more special than I thought lol
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