Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

the right) bybwe obtain

a†b=(a∗ 1 a∗ 2 ···a∗N)











b 1

b 2

..

.

bN











=

∑N

i=1

a∗ibi, (8.42)

which is the expression for the inner product〈a|b〉in that basis. We note that for

real vectors (8.42) reduces toaTb=

∑N

i=1aibi.

If the basiseiisnotorthonormal, so that, in general,

〈ei|ej〉=Gij=δij,

then, from (8.18), the scalar product ofaandbin terms of their components with

respect to this basis is given by

〈a|b〉=

∑N

i=1

∑N

j=1

a∗iGijbj=a†Gb,

whereGis theN×Nmatrix with elementsGij.

8.8 The trace of a matrix

For a given matrixA, in the previous two sections we have considered various

other matrices that can be derived from it. However, sometimes one wishes to

derive a single number from a matrix. The simplest example is thetrace(orspur)

of a square matrix, which is denoted by TrA. This quantity is defined as the sum

of the diagonal elements of the matrix,

TrA=A 11 +A 22 +···+ANN=

∑N

i=1

Aii. (8.43)

It is clear that taking the trace is a linear operation so that, for example,

Tr(A±B)=TrA±TrB.

A very useful property of traces is that the trace of the product of two matrices

is independent of the order of their multiplication; this results holds whether or

not the matrices commute and is proved as follows:

TrAB=

∑N

i=1

(AB)ii=

∑N

i=1

∑N

j=1

AijBji=

∑N

i=1

∑N

j=1

BjiAij=

∑N

j=1

(BA)jj=TrBA.

(8.44)

The result can be extended to the product of several matrices. For example, from

(8.44), we immediately find

TrABC=TrBCA=TrCAB,