From page 336 of "Numerical recipes in Fortran", 1989
Symmetric \equiv A = A^T \quad \quad a_{ij} = a_{ji} \\
Hermitian = self-adjoint \equiv A = A^+ \quad \quad a_{ij} = a_{ji}^* \\
orthogonal \equiv A^T A = A A^T = 1 \\
unitary (conservation of flux) \equiv A^+ A = A^+ A = 1 \\
normal \equiv A^+ A = A^+ A
if A \in \Re, then (Hermitian = symmetric) and (unitary = orthogonal)