nLab conjugate transpose matrix

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

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

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.

Related concepts

• transpose matrix

• hermitian matrix

• bilinear form