The adjoint A* of a matrix A is its hermitian conjugate. That is, entry (i,j) of the adjoint A* is the complex conjugate of entry (j,i) of matrix A. If matrix A only contains real entries, then its adjoint is its transpose. If A = A* (which can only happen if A is square), then A is said to be hermitian. If an hermitian matrix has only real elements, then it is said to be a symmetric. Take care; the adjoint of a matrix is not the adjoint matrix...