Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.15 CHANGE OF BASIS AND SIMILARITY TRANSFORMATIONS

and representxin this basis by the column matrix

x=(x 1 x 2 ···xn)T,

having componentsxi. We now consider how these components change as a result

of a prescribed change of basis. Let us introduce a new basise′i,i=1, 2 ,...,N,

which is related to the old basis by

e′j=

∑N

i=1

Sijei, (8.90)

the coefficientSijbeing theith component ofe′jwith respect to the old (unprimed)

basis. For an arbitrary vectorxit follows that

x=

∑N

i=1

xiei=

∑N

j=1

x′je′j=

∑N

j=1

x′j

∑N

i=1

Sijei.

From this we derive the relationship between the components ofxin the two

coordinate systems as

xi=

∑N

j=1

Sijx′j,

which we can write in matrix form as

x=Sx′ (8.91)

whereSis thetransformation matrixassociated with the change of basis.

Furthermore, since the vectorse′jare linearly independent, the matrixSis

non-singular and so possesses an inverseS−^1. Multiplying (8.91) on the left by

S−^1 we find

x′=S−^1 x, (8.92)

which relates the components ofxin the new basis to those in the old basis.

Comparing (8.92) and (8.90) we note that the components ofxtransform inversely

to the way in which the basis vectorseithemselves transform. This has to be so,

as the vectorxitself must remain unchanged.

We may also find the transformation law for the components of a linear

operator under the same change of basis. Now, the operator equationy=Ax

(which is basis independent) can be written as a matrix equation in each of the

two bases as

y=Ax, y′=A′x′. (8.93)

But, using (8.91), we may rewrite the first equation as

Sy′=ASx′ ⇒ y′=S−^1 ASx′.