Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

kirbykirby

  • 3 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?

  • This Question is Closed
  1. kirbykirby
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    See here for a little more information http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse

  4. kirbykirby
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    o_o interesting. Hmm lol this Hat matrix is a lot more special than I thought lol

  5. KingGeorge
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Same here.

  6. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy