| Relations | Name | matrix elements |
| A = A^{T} | symmetric | a_{ij} = a_{ji} |
| A = \left (A^{T} \right )^{-1} | real orthogonal | \sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij} |
| A = A^{ * } | real matrix | a_{ij} = a_{ij}^{ * } |
| A = A^{\dagger} | hermitian | a_{ij} = a_{ji}^{ * } |
| A = \left (A^{\dagger} \right )^{-1} | unitary | \sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij} |