KnifeMissile

I believe that, in order for a matrix to be orthogonally diagonal, it must have no zero columns (diagonal matrices may have zero columns). If so, then it's a simple matter to prove that, for every orthogonally diagonal matrix, there exists a multiplicative inverse and that inverse is also a orthogonally diagonal.

Knowing this, if (1/P)AP = D, an orthogonally diagonal matrix, then 1/( (1/P)AP ) = (1/P)(1/A)P = 1/D and, because 1/D is also orthogonally diagonal, 1/A must also be orthogonally diagonalizable.
QED.


...edited for correctness...