The trace of a tensor is defined as the sum of the diagonal elements:
tr{I}≡k∑Ikk
Show, by performing a similarity transformation, that the trace is an invariant quantity. In other words, show that
tr{I}=tr{I′}
where {I} is the tensor in one coordinate system and {I}′ is the tensor in a coordinate system rotated with respect to the first system. Verify this result for the different forms of the inertia tensor for a cube given in several examples in the text.