Let ij , with inverse ij , be the components of a p p symmetric

Let ω

ij

, with inverse ωij

, be the components of a p × p symmetric matrix of rank p. Show that

γ

rs,ij = ω

riω

sj + ω

rjω

si

, regarded as a p 2 × p 2 matrix with rows indexed by (r, s) and columns by (i, j), is symmetric with rank p(p+l)/2. Show also that ωriωsj

/2 is a generalized inverse. Find the Moore-Penrose generalized inverse.

1.13 Consider the linear mapping from R p to itself given by X r = ω

r iXi where ω

r i

is nonsingular. Show that, under simultaneous change of coordinates Y

r = a r

iXi

, Y r = a r

iX i

,

ω

r i

transforms as a mixed tensor. By comparing the volume of a set, B say, in the X coordinate system with the volume of the transformed set, B̅, interpret the determinant of ω

r i

as an invariant. Give similar interpretations of the remaining p − 1 invariants, e.g. in terms of surface area and so on.