Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 10 / a / ii

A symmetric matrix has $A^T=A$, without conjugation. One invertible example with nonreal off-diagonal entries isIt is symmetric but not Hermitian, illustrating the importance of the distinction over $\mathbb{C}$.

A Hermitian matrix has $A^{\ast }=A$. ChooseIts off-diagonal entries are complex conjugates and its diagonal entries are real, so the defining condition holds. It is also invertible.

A unitary matrix has $A^{\ast }A=I$. For example,This example is invertible, has nonreal off-diagonal entries, and preserves the standard Hermitian inner product. Although it is also symmetric, that additional property is not required for unitarity.