I've seen a lot of discussion going the other way, but I'm considering the fact that if a matrix A is orthogonally diagonalizable then $A=PDP^{-1}=PDP^{T}$. And because the RHS is invertible so is A?
2026-04-06 01:07:52.1775437672
Is every orthogonally diagonalizable matrix invertible?
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Take a diagonal matrix $D$ with some zero on the diagonal. Then it is not invertible. Then take whatever orthogonal and invertible matrix you want $P$.
Then $A = PDP^{T}$ is clearly not invertible (since it has the same rank as $D$), but orthogonally diagonalizable.
$0$ is a trivial example.