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 one year 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?
 one year 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
 one year 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
 one year 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
 one year 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
 one year 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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