The complex and Hermitian conjugates of a matrix

MATRICES AND VECTOR SPACES
Find the transpose of the matrix
A=
3
0
1
4
2
1
.
By interchanging the rows and columns of A we immediately obtain


3 0
T

1 4 . A =
2 1
It is obvious that if A is an M × N matrix then its transpose AT is a N × M
matrix. As mentioned in section 8.3, the transpose of a column matrix is a
row matrix and vice versa. An important use of column and row matrices is
in the representation of the inner product of two real vectors in terms of their
components in a given basis. This notion is discussed fully in the next section,
where it is extended to complex vectors.
The transpose of the product of two matrices, (AB)T , is given by the product
of their transposes taken in the reverse order, i.e.
(AB)T = BT AT .
(8.39)
This is proved as follows:
(AB)Tij = (AB)ji =
Ajk Bki
k
=
(AT )kj (BT )ik =
k
(BT )ik (AT )kj = (BT AT )ij ,
k
and the proof can be extended to the product of several matrices to give
(ABC · · · G)T = GT · · · CT BT AT .
8.7 The complex and Hermitian conjugates of a matrix
Two further matrices that can be derived from a given general M × N matrix
are the complex conjugate, denoted by A∗ , and the Hermitian conjugate, denoted
by A† .
The complex conjugate of a matrix A is the matrix obtained by taking the
complex conjugate of each of the elements of A, i.e.
(A∗ )ij = (Aij )∗ .
Obviously if a matrix is real (i.e. it contains only real elements) then A∗ = A.
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8.7 THE COMPLEX AND HERMITIAN CONJUGATES OF A MATRIX
Find the complex conjugate of the matrix
1
A=
1+i
2
1
3i
0
.
By taking the complex conjugate of each element we obtain immediately
1
2 −3i
.
A∗ =
1−i 1
0
The Hermitian conjugate, or adjoint, of a matrix A is the transpose of its
complex conjugate, or equivalently, the complex conjugate of its transpose, i.e.
A† = (A∗ )T = (AT )∗ .
We note that if A is real (and so A∗ = A) then A† = 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† .
Find the Hermitian conjugate of the matrix
1
A=
1+i
2
1
3i
0
(8.40)
.
Taking the complex conjugate of A and then forming the transpose we find


1
1−i
1 .
A† =  2
−3i
0
We obtain the same result, of course, if we first take the transpose of A and then take the
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 vectors a and b may be represented by the column matrices



a=

a1
a2
..
.






and
aN


b=

b1
b2
..
.



.

(8.41)
bN
Taking the Hermitian conjugate of a, to give a row matrix, and multiplying (on
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