26.2 Change of basis
In the second of these the dummy index shared by both terms on the left-hand
side (namelyj) has been replaced by the free index carried by the Kronecker delta
(namelyk), and the delta symbol has disappeared. In matrix language, (26.1) can
be written asAI=A,whereAis the matrix with elementsaijandIis the unit
matrix having the same dimensions asA.
In some expressions we may use the Kronecker delta to replace indices in anumber of different ways, e.g.
aijbjkδki=aijbji or akjbjk,where the two expressions on the RHS are totally equivalent to one another.
26.2 Change of basisIn chapter 8 some attention was given to the subject of changing the basis set (or
coordinate system) in a vector space and it was shown that, under such a change,
different types of quantity behave in different ways. These results are given in
section 8.15, but are summarised below for convenience, using the summation
convention. Although throughout this section we will remind the reader that we
are using this convention, it will simply be assumed in the remainder of the
chapter.
If we introduce a set of basis vectorse 1 ,e 2 ,e 3 into our familiar three-dimensional(vector) space, then we can describe any vectorxin terms of its components
x 1 ,x 2 ,x 3 with respect to this basis:
x=x 1 e 1 +x 2 e 2 +x 3 e 3 =xiei,where we have used the summation convention to write the sum in a more
compact form. If we now introduce a new basise′ 1 ,e′ 2 ,e′ 3 related to the old one by
e′j=Sijei (sum overi), (26.2)where the coefficientSijis theith component of the vectore′jwith respect to the
unprimed basis, then we may writexwith respect to the new basis as
x=x′ 1 e′ 1 +x′ 2 e′ 2 +x′ 3 e′ 3 =x′ie′i (sum overi).If we denote the matrix with elementsSijbyS, then the componentsx′iandxiin the two bases are related by
x′i=(S−^1 )ijxj (sum overj),where, using the summation convention, there is an implicit sum overjfrom
j=1toj= 3. In the special case where the transformation is a rotation of the
coordinate axes, the transformation matrixSis orthogonal and we have
x′i=(ST)ijxj=Sjixj (sum overj). (26.3)