An orthogonal matrix

Question

Let $U$ be an orthogonal matrix with real spectra. Is it true that since the spectrum of $U$ is $\{-1,1\}$, so $U^{2}$ is similar to the identity matrix?

Answer 1

Three things:

• The only matrix similar to the identity matrix is the identity matrix. For any invertible $P$, $PIP^{-1} = I$.
• If $U$ is any diagonalizable matrix with spectrum $\{1,-1\}$, then $U^2 = I$. One way to see this is by noting that $U$ would have minimal polynomial $x^2 - 1$.
• All orthogonal matrices are diagonalizable over $\Bbb C$ by the spectral theorem for normal matrices

All together: if $U$ is a real orthogonal matrix with a real spectrum, then $U$ is diagonalizable with spectrum (a subset of) $\{-1,1\}$ and $U^2 = I$.