I read this a lot of times, but I can't seem to prove it. How does a positive definite matrix $A$ decompose to $QQ^T$?

  • 1
    $\begingroup$ Look at the associated quadratic form, and repeatedly "complete the square". $\endgroup$ – Lord Shark the Unknown May 10 '18 at 6:43
  • $\begingroup$ Not every positive definite matrix is orthogonal. $\endgroup$ – Sean Roberson May 10 '18 at 6:44
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    $\begingroup$ Use the spectral decomposition and then take the square roots of the (positive) eigenvalues. Take a look at this. $\endgroup$ – Rodrigo de Azevedo May 10 '18 at 6:56
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    $\begingroup$ You may also repeatedly "remove squares" (as opposed to completing square). This results in Cholesky decomposition. $\endgroup$ – user1551 May 10 '18 at 7:01

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