| 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 | \( \sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij} \) |