Find a unitary matrix which is not orthogonal, and find an orthogonal matrix which is not unitary.
Solution
1. A unitary matrix need not be orthogonal.
Take the \(1\times1\) complex matrix \[ U=(i). \] Its conjugate transpose is \(U^*=(-i)\), so \[ U^*U=(-i)i=1. \] Thus \(U\) is unitary. But orthogonality uses transpose rather than conjugate transpose, and \[ U^tU=i^2=-1\ne1. \] Hence \(U\) is not orthogonal.
2. An orthogonal complex matrix need not be unitary.
Consider \[ Q= \begin{pmatrix} \sqrt2&-i\\ i&\sqrt2 \end{pmatrix}. \] Its two columns have transpose-dot-products \[ 2+i^2=1, \qquad (-i)^2+2=1, \] and their mutual transpose-dot-product is \[ \sqrt2(-i)+i\sqrt2=0. \] Therefore \[ Q^tQ=I, \] so \(Q\) is orthogonal.
On the other hand, the Hermitian squared norm of its first column is \[ |\sqrt2|^2+|i|^2=2+1=3, \] so its columns are not orthonormal for the Hermitian inner product. Thus \[ Q^*Q\ne I, \] and \(Q\) is not unitary.