Mathematical Methods for Physics and Engineering - Matematica.NET

8.7 THE COMPLEX AND HERMITIAN CONJUGATES OF A MATRIX◮Find the complex conjugate of the matrix(1 2 3iA =1+i 1 0).By taking the complex conjugate of each element we obtain immediately( )1 2 −3iA ∗ =. ◭1 − i 1 0The Hermitian conjugate, or adjoint, of a matrix A is the transpose of itscomplex conjugate, or equivalently, the complex conjugate of its transpose, i.e.A † =(A ∗ ) T =(A T ) ∗ .We note that if A is real (and so A ∗ = A) thenA † = A T , and taking the Hermitianconjugate is equivalent to taking the transpose. Following the previous line ofargument for the transpose of the product of several matrices, the Hermitianconjugate of such a product can be shown to be given by(AB ···G) † = G † ···B † A † . (8.40)◮Find the Hermitian conjugate of the matrix(1 2 3iA =1+i 1 0).Taking the complex conjugate of A and then forming the transpose we find⎛A † = ⎝ 1 1− i⎞2 1 ⎠ .−3i 0We obtain the same result, of course, if we first take the transpose of A andthentakethecomplex conjugate. ◭An important use of the Hermitian conjugate (or transpose in the real case)is in connection with the inner product of two vectors. Suppose that in a givenorthonormal basis the vectors a and b may be represented by the column matrices⎛a = ⎜⎝a 1a 2.a N⎞⎟⎠⎛and b = ⎜⎝b 1b 2.b N⎞⎟⎠ . (8.41)Taking the Hermitian conjugate of a, to give a row matrix, and multiplying (on257