Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 6 / Solution

A real square orthogonal matrix satisfies $Q^{T}Q=I$. A complex square unitary matrix satisfies $U^{†}U=I$, where $†$ denotes conjugate transpose. A Hermitian matrix satisfies $H^{†}=H$. The distinction between transpose and conjugate transpose matters for imaginary entries.

For the three displayed matrices, in their source order:

For the first matrix, $A^{†}=-A$ and $A^2=-I$, so $A^{†}A=I$ but $A^{†}\ne A$. It also fails $A^{T}A=I$, even if complex orthogonality is considered. The second matrix is real diagonal with entries $1,-1$: its transpose, conjugate transpose and inverse all equal itself. In the third matrix the imaginary off-diagonal entries are conjugate pairs, establishing the Hermitian matrix condition. Its first column has squared norm $1+3=4$, so it is not a unitary matrix; it is not real orthogonal, and the complex bilinear first-column square is $1-3=-2\ne 1$ as well.