Given some matrix $A$, how do I find a diagonal matrix $D$ and an orthogonal $P$ so that $D = P^t A P$? I know how it's done with a regular $P$ which is invertible, how is it different with $P$ is orthogonal?

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    $\begingroup$ This isn't always possible. Even if $A$ is assumed to be diagonalizable, it's not always possible. But if it possible, you just have to find $P$ in the usual way and then you apply G.S. to its columns. $\endgroup$ – Git Gud Apr 25 '14 at 14:02
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    $\begingroup$ You can do this if and only if $A$ is (real) symmetric. $\endgroup$ – Brandon Carter Apr 25 '14 at 14:08
  • $\begingroup$ You can also do something very similar if $A$ is (complex) Hermitian (i.e., $\overline{A}^t = A$), in which case one can find a unitary matrix $P$ (i.e., $P^{-1} = \overline{P}^t$) with $D = \overline{P}^t A P$ a diagonal matrix. (here, $\overline{a+ib} = a-ib$ is complex conjugation) $\endgroup$ – Nicholas Stull Apr 25 '14 at 15:05

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