What is the trace of the second-Rank tensor?

[PRB147]

In Lorentz group in QFT, why the trace of the symmetric second-Rank tensor $$S^{\mu\nu}$$ is defined as follows?
$$S=g_{\mu\nu}S^{\mu\nu}$$.
Is it just a definition or the genuine trace of the second-Rank tensor, and why?

 

[cristo]

What's the trace of a matrix?

 

[PRB147]

my opinion,The trace of $$S^{ \mu\nu}$$ should be $$S^{ 00}+S^{11}+S^{22}+S^{33}$$
, i.e. the sum over the diagonal elements.
Now the metric tensor $$g_{ \mu\nu}$$ is taken into account, the trace becomes
$$S^{ 00}-S^{11}-S^{22}-S^{33}$$.
The problem is that I don't know why

 

[genneth]

This seems to be a common point of misunderstanding:

Matrices properly correspond to tensors with one index up and one index down.

The fact that books often just write the metric with all up or all down as a matrix is just confusing.

 

[Meir Achuz]

The basic definition of the trace of a matrix is simply the sum of its diagonal elements.
However, in order to make the trace invariant under a generalized rotation, the metric is included.

 

[PRB147]

Thank Cristo, genneth, and clem.
You are correct, it is described in general relativity.