Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.3.
The trace of a rank-2 tensor is given by the contraction . In matrix terminology, it is the sum of the diagonal elements. If we start with a contravariant tensor , then we can calculate the trace as follows:
That is, we first lower the second index, then contract the top and bottom indices.
Since the trace contains no free index, it should be a scalar, which means it should be invariant. We can prove this by doing the transformation.