Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESthe right) by b we obtain⎛a † b =(a ∗ 1 a ∗ 2 ··· a ∗ N) ⎜⎝b 1b 2.b N⎞N⎟⎠ = ∑a ∗ i b i , (8.42)which is the expression for the inner product 〈a|b〉 in that basis. We note that forreal vectors (8.42) reduces to a T b = ∑ Ni=1 a ib i .If the basis e i is not orthonormal, so that, in general,〈e i |e j 〉 = G ij ≠ δ ij ,then, from (8.18), the scalar product of a and b in terms of their components withrespect to this basis is given by〈a|b〉 =N∑i=1 j=1i=1N∑a ∗ i G ij b j = a † Gb,where G is the N × N matrix with elements G ij .8.8 The trace of a matrixFor a given matrix A, in the previous two sections we have considered variousother matrices that can be derived from it. However, sometimes one wishes toderive a single number from a matrix. The simplest example is the trace (or spur)of a square matrix, which is denoted by Tr A. This quantity is defined as the sumof the diagonal elements of the matrix,Tr A = A 11 + A 22 + ···+ A NN =N∑A ii . (8.43)It is clear that taking the trace is a linear operation so that, for example,Tr(A ± B) =TrA ± Tr B.A very useful property of traces is that the trace of the product of two matricesis independent of the order of their multiplication; this results holds whether ornot the matrices commute and is proved as follows:N∑N∑ N∑N∑ N∑N∑Tr AB = (AB) ii = A ij B ji = B ji A ij =i=1i=1 j=1i=1 j=1i=1j=1(BA) jj =TrBA.(8.44)The result can be extended to the product of several matrices. For example, from(8.44), we immediately findTr ABC =TrBCA =TrCAB,258