Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

|expA|. Moreover, by choosing the similarity transformation so that it diagonalisesA,we
haveA′=diag(λ 1 ,λ 2 ,...,λN), and so

|expA|=|expA′|=|exp[diag(λ 1 ,λ 2 ,...,λN)]|=|diag(expλ 1 ,expλ 2 ,...,expλN)|=

∏N

i=1

expλi.

Rewriting the final product of exponentials of the eigenvalues as the exponential of the
sum of the eigenvalues, we find

|expA|=

∏N

i=1

expλi=exp

(N

∑

i=1

λi

)

=exp(TrA),

which gives the trace formula (8.104).

8.17 Quadratic and Hermitian forms

Let us now introduce the concept of quadratic forms (and their complex ana-

logues, Hermitian forms). A quadratic formQis a scalar function of a real vector

xgiven by

Q(x)=〈x|Ax〉, (8.105)

for some real linear operatorA. In any given basis (coordinate system) we can

write (8.105) in matrix form as

Q(x)=xTAx, (8.106)

whereAis a real matrix. In fact, as will be explained below, we need only consider

the case whereAis symmetric, i.e.A=AT. As an example in a three-dimensional

space,

Q=xTAx=

(

x 1 x 2 x 3

)





11 3

11 − 3

3 − 3 − 3









x 1

x 2

x 3





=x^21 +x^22 − 3 x^23 +2x 1 x 2 +6x 1 x 3 − 6 x 2 x 3. (8.107)

It is reasonable to ask whether a quadratic formQ=xTMx,whereMis any

(possibly non-symmetric) real square matrix, is a more general definition. That

this is not the case may be seen by expressingMin terms of a symmetric matrix

A=^12 (M+MT) and an antisymmetric matrixB=^12 (M−MT) such thatM=A+B.

We then have

Q=xTMx=xTAx+xTBx. (8.108)

However,Qis a scalar quantity and so

Q=QT=(xTAx)T+(xTBx)T=xTATx+xTBTx=xTAx−xTBx.

(8.109)

Comparing (8.108) and (8.109) shows thatxTBx= 0, and hencexTMx=xTAx,