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kirbykirby

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

  2. KingGeorge
    • 2 years ago
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    Took a little research, but you're correct. However, if \(X\) does not have an inverse, this does not result in the identity.

  3. KingGeorge
    • 2 years ago
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    See here for a little more information http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse

  4. kirbykirby
    • 2 years ago
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    o_o interesting. Hmm lol this Hat matrix is a lot more special than I thought lol

  5. KingGeorge
    • 2 years ago
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    Same here.

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