Change of basis

26.2 CHANGE OF BASIS
In the second of these the dummy index shared by both terms on the left-hand
side (namely j) has been replaced by the free index carried by the Kronecker delta
(namely k), and the delta symbol has disappeared. In matrix language, (26.1) can
be written as AI = A, where A is the matrix with elements aij and I is the unit
matrix having the same dimensions as A.
In some expressions we may use the Kronecker delta to replace indices in a
number of different ways, e.g.
aij bjk δki = aij bji
or
akj bjk ,
where the two expressions on the RHS are totally equivalent to one another.
26.2 Change of basis
In 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 vectors e1 , e2 , e3 into our familiar three-dimensional
(vector) space, then we can describe any vector x in terms of its components
x1 , x2 , x3 with respect to this basis:
x = x1 e1 + x2 e2 + x3 e3 = xi ei ,
where we have used the summation convention to write the sum in a more
compact form. If we now introduce a new basis e1 , e2 , e3 related to the old one by
ej = Sij ei
(sum over i),
(26.2)
where the coefficient Sij is the ith component of the vector ej with respect to the
unprimed basis, then we may write x with respect to the new basis as
x = x1 e1 + x2 e2 + x3 e3 = xi ei
(sum over i).
If we denote the matrix with elements Sij by S, then the components xi and xi
in the two bases are related by
xi = (S−1 )ij xj
(sum over j),
where, using the summation convention, there is an implicit sum over j from
j = 1 to j = 3. In the special case where the transformation is a rotation of the
coordinate axes, the transformation matrix S is orthogonal and we have
xi = (ST )ij xj = Sji xj
929
(sum over j).
(26.3)