Can orthogonal matrix be complex?

Question

A matrix $Q$ is orthogonal if $Q^TQ=I$. My question is can $Q$ be complex? Can anyone help show an example, or provide a proof that $Q$ has to be real? Thanks.

Answer 1

There's no reason why $Q$ has to be real. For instance, the matrix $$ Q=\begin{bmatrix}i&\sqrt{2}\\-\sqrt{2}&i\end{bmatrix} $$ is orthogonal.

However, unitary matrices over $\mathbb{C}$ are really the natural generalization of orthogonal matrices over $\mathbb{R}$. For instance, unitary matrices preserve the inner product of two vectors, and the group of $n\times n$ unitary matrices is compact.