TITLE: Is there an orthogonal matrix that is not unitary?
QUESTION [5 upvotes]: I could find a example of a unitary matrix such that is not orthogonal, thats simple in $\mathbb{C}$, but for this exercise of a orthogonal that is not unitary i realize that is possible just on $\mathbb{C}$ because all orthogonal matrix on $\mathbb{R}$ is unitary, so anyone have a exemple of this case?

REPLY [2 votes]: Late remark. In general, if $K$ is complex skew-symmetric, then $Q=e^{zK}$ is complex orthogonal for every $z\in\mathbb C$. When $K\ne0$, since $e^{zK}$ is holomorphic and its power series expansion has some nonzero high-order terms, $e^{zK}$ is not a constant function. Therefore it cannot be real all the time (otherwise it will fail to satisfy Cauchy-Riemann equations). Pick a $z$ such that $Q=e^{zK}$ is not real. Then $Q$ is not unitary, because all unitary orthogonal matrices are real.