8.7 THE COMPLEX AND HERMITIAN CONJUGATES OF A MATRIX
Find the complex conjugate of the matrixA=(
123 i
1+i 10)
.
By taking the complex conjugate of each element we obtain immediately
A∗=
(
12 − 3 i
1 −i 10)
.
The Hermitian conjugate, oradjoint,ofamatrixAis the transpose of itscomplex conjugate, or equivalently, the complex conjugate of its transpose, i.e.
A†=(A∗)T=(AT)∗.We note that ifAis real (and soA∗=A)thenA†=AT, and taking the Hermitian
conjugate is equivalent to taking the transpose. Following the previous line of
argument for the transpose of the product of several matrices, the Hermitian
conjugate of such a product can be shown to be given by
(AB···G)†=G†···B†A†. (8.40)Find the Hermitian conjugate of the matrixA=(
123 i
1+i 10)
.
Taking the complex conjugate ofAand then forming the transpose we find
A†=
11 −i
21
− 3 i 0
.
We obtain the same result, of course, if we first take the transpose ofAandthentakethe
complex 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 given
orthonormal basis the vectorsaandbmay be represented by the column matrices
a=
a 1
a 2
..
.
aN
and b=
b 1
b 2
..
.
bN
. (8.41)
Taking the Hermitian conjugate ofa, to give a row matrix, and multiplying (on