The adjoint of a matrix is the transpose of the conjugate of the matrix. The conjugate of the matrix can be determined by swapping every element in the matrix with its complex conjugate. The transpose can be found by swapping every element a(ij) of the matrix with the element a(ji).
In all separable Hilbert spaces, the transpose of the conjugate of a matrix is identical to the conjugate of the transpose of the matrix, so it does not matter in which order these operations are performed when calculating the adjoint. Some matrices are their own adjoint. These matrices are called self-adjoint or Hermitian matrices.