info prev up next book cdrom email home

Adjoint Matrix

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

{\hbox{\sf A}}^\dagger \equiv ({\hbox{\sf A}}^{\rm T})^*,
\end{displaymath} (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}\,.
\end{displaymath} (2)

Using the property of transpose products that
$\displaystyle {[}(ab)^{\rm T}]^*_{ij}$ $\textstyle =$ $\displaystyle (b^{\rm T} a^{\rm T})^*_{ij} = (b^{\rm T}_{ik} a^{\rm T}_{kj})^* = (b^{\rm T})^*_{ik} (a^{\rm T})^*_{kj}$  
  $\textstyle =$ $\displaystyle b^\dagger_{ik} a^\dagger_{kj} = (b^\dagger a^\dagger)_{ij}\,,$ (3)

it follows that
({\hbox{\sf A}}{\hbox{\sf B}})^\dagger = {\hbox{\sf B}}^\dagger{\hbox{\sf A}}^\dagger.
\end{displaymath} (4)

© 1996-9 Eric W. Weisstein