Consider the decomposition ij = r i s j rs of the symmetric covariant tensor

Consider the decomposition ωij = τ

r i

τ

s j

ϵrs of the symmetric covariant tensor ωij

, where the notation is that used in (1.14). Define

ω

+

ij = τ

r i

τ

s j

|ϵrs|.

Show that ω

+

ij is a covariant tensor and that the scalar s = ω

ijω

+

ij is independent of the choice of τ

r i

and also of the choice of generalized inverse ω

ij

. Show that s is the signature of ωij

.