Mathematical Methods for Physics and Engineering - Matematica.NET

26.2 CHANGE OF BASISIn the second of these the dummy index shared by both terms on the left-handside (namely j) has been replaced by the free index carried by the Kronecker delta(namely k), and the delta symbol has disappeared.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) canbe written as AI = A, whereA is the matrix with elements a ij and I is the unitmatrix having the same dimensions as A.In some expressions we may use the Kronecker delta to replace indices in anumber of different ways, e.g.a ij b jk δ ki = a ij b ji or a kj b jk ,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 (orcoordinate 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 insection 8.15, but are summarised below for convenience, using the summationconvention. Although throughout this section we will remind the reader that weare using this convention, it will simply be assumed in the remainder of thechapter.If we introduce a set of basis vectors e 1 , e 2 , e 3 into our familiar three-dimensional(vector) space, then we can describe any vector x in terms of its componentsx 1 ,x 2 ,x 3 with respect to this basis:x = x 1 e 1 + x 2 e 2 + x 3 e 3 = x i e i ,where we have used the summation convention to write the sum in a morecompact form. If we now introduce a new basis e ′ 1 , e′ 2 , e′ 3 related to the old one bye ′ j = S ij e i (sum over i), (26.2)where the coefficient S ij is the ith component of the vector e ′ j with respect to theunprimed basis, then we may write x with respect to the new basis asx = x ′ 1e ′ 1 + x ′ 2e ′ 2 + x ′ 3e ′ 3 = x ′ ie ′ i(sum over i).If we denote the matrix with elements S ij by S, then the components x ′ i and x iin the two bases are related byx ′ i =(S −1 ) ij x j(sum over j),where, using the summation convention, there is an implicit sum over j fromj =1toj = 3. In the special case where the transformation is a rotation of thecoordinate axes, the transformation matrix S is orthogonal and we havex ′ i =(S T ) ij x j = S ji x j (sum over j). (26.3)929