1.14 (

) Show that an arbitrary square matrix with elements wij can be written in the form wij = wS ij + wA ij where wS ij and wA ij are symmetric and anti-symmetric matrices, respectively, satisfying wS ij = wS ji and wA ij = −wA ji for all i and j. Now consider the second order term in a higher order polynomial in D dimensions, given by

D i=1

D j=1 wijxixj . (1.131)

Show that

D i=1

D j=1 wijxixj =

D i=1

D j=1 wS ijxixj (1.132)

so that the contribution from the anti-symmetric matrix vanishes. We therefore see that, without loss of generality, the matrix of coefficients wij can be chosen to be symmetric, and so not all of the D2 elements of this matrix can be chosen independently.

Show that the number of independent parameters in the matrix wS ij is given by D(D + 1)/2.