Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

Find the transpose of the matrix

A=

(

312

041

)

.

By interchanging the rows and columns ofAwe immediately obtain

AT=





30

14

21



.

It is obvious that ifAis anM×Nmatrix then its transposeATis aN×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=BTAT. (8.39)

This is proved as follows:

(AB)Tij=(AB)ji=

∑

k

AjkBki

=

∑

k

(AT)kj(BT)ik=

∑

k

(BT)ik(AT)kj=(BTAT)ij,

and the proof can be extended to the product of several matrices to give

(ABC···G)T=GT···CTBTAT.

8.7 The complex and Hermitian conjugates of a matrix

Two further matrices that can be derived from a given generalM×Nmatrix

are thecomplex conjugate, denoted byA∗,andtheHermitian conjugate, denoted

byA†.

The complex conjugate of a matrixAis the matrix obtained by taking the

complex conjugate of each of the elements ofA,i.e.

(A∗)ij=(Aij)∗.

Obviously if a matrix isreal(i.e. it contains only real elements) thenA∗=A.