TITLE: Geometric interpretation of trace
QUESTION [329 upvotes]: This afternoon I was speaking with some graduate students in the department and we came to the following quandary;

Is there a geometric interpretation of the trace of a matrix?

This question should make fair sense because trace is coordinate independent.
A few other comments. We were hoping for something like: 
"determinant is the volume of the parallelepiped spanned by column vectors."
This is nice because it captures the geometry simply, and it holds for any old set of vectors over $\mathbb{R}^n$.
The divergence application of trace is somewhat interesting, but again, not really what we are looking for.
Also, after looking at the wiki entry, I don't get it. This then requires a matrix function, and I still don't really see the relationship.
One last thing that we came up with; the trace of a matrix is the same as the sum of the eigenvalues. Since eigenvalues can be seen as the eccentricity of ellipse, trace may correspond geometrically to this. But we could not make sense of this.

REPLY [3 votes]: For me trace of a matrix always was analogous to the real part of a complex number. As such, I consider trace divided by the matrix' rank as the "scalar part" of the matrix.
There are other analogies:

For analytic functions trace is analog of the value of the function at zero or other central point of their domain (which is the constant term in the functions' Taylor expansion).

For divergent integrals and series, trace is analog of the regularized value.

For vectors in spacetime, it is analog of the time component of the vector.