Give an example of an (complex) orthogonal matrix that isn't unitary and vice versa.

Question

Can you give the simplest example you could think of please.

I'm kind of collecting such awkward examples at the moment and have collected quite a few concerning e.g. normal matrices that aren't unitary or hermitian etc. but I'm having trouble getting examples as in the question.

Thanks in advance.

Answer 1

I only know the term "orthogonal matrix" applied to real matrices; I assume that in applying it to complex matrices you intend to require $O^\top O=1$ as for real matrices.

A matrix that is unitary but not orthogonal is $\pmatrix{\mathrm i}$.

A matric that is orthogonal but not unitary is $\displaystyle\frac1{\sqrt3}\pmatrix{\mathrm 2&\mathrm i\\\mathrm i&-\mathrm 2}$.