Adjoint Matrix

The adjoint matrix, sometimes also called the Adjugate Matrix, is defined by

$${\hbox{\sf A}}^\dagger \equiv ({\hbox{\sf A}}^{\rm T})^*,$$ (1)

where the Adjoint Operator is denoted ${}^\dagger$ and ${}^{\rm T}$ denotes the Transpose. If a Matrix is Self-Adjoint, it is said to be Hermitian. The adjoint matrix of a Matrix product is given by

$${(ab)^\dagger}_{ij} \equiv [(ab)^{\rm T}]^*_{ij}\,.$$ (2)

Using the property of transpose products that

$${[}(ab)^{\rm T}]^*_{ij} = (b^{\rm T} a^{\rm T})^*_{ij} = (b^{\rm T}_{ik} a^{\rm T}_{kj})^* = (b^{\rm T})^*_{ik} (a^{\rm T})^*_{kj}$$

$$= b^\dagger_{ik} a^\dagger_{kj} = (b^\dagger a^\dagger)_{ij}\,,$$ (3)

it follows that

$$({\hbox{\sf A}}{\hbox{\sf B}})^\dagger = {\hbox{\sf B}}^\dagger{\hbox{\sf A}}^\dagger.$$ (4)