I am working on a problem where I am just given the characteristic polynomial of some 3x3 matrix $C$ and am asked to show that it is similar to an orthogonal matrix.
So I solved for the roots of the polynomial and found 3 distinct eigenvalues: 1 real, 2 complex.
So $C$ is similar to a diagonal matrix $D$, with the eigenvalues on the diagonal.
I computed $DD^*$, which happened to equal $D^*D$, which equaled to $I$. So, I know that $C$ is similar to a unitary matrix $D$.
However the problem statement asks to show the similarity to an orthogonal matrix, so I wonder whether the matrix $D$ suffices, or whether I have to now find another matrix, from knowing $D$.
Any ideas are welcome.