nLab
conjugate transpose matrix

If A=(a ij)A = (a_{i j}) is a matrix with coefficients in a star algebra (such as the complex numbers under complex conjugation), then its conjugate transpose A A^\dagger is the matrix A (a ji *)A^\dagger \coloneqq (a^\ast_{j i}), hence the composite of passing to the transpose matrix and applying the star-operation

A (A t) *=(A *) t A^\dagger \coloneqq \left(A^t\right)^\ast = \left(A^\ast\right)^t

Identifying matrices with linear maps and with respect to the standard inner product this operation represents passing to the adjoint operator. Therefore one speaks also of adjoint matrices.

Last revised on August 1, 2018 at 08:15:13. See the history of this page for a list of all contributions to it.