Question

What is the trace of a transformation matrix? How is this related to the character of a transformation matrix?

Answer 1

The Trace of a square matrix A is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A.

Trace of a linear operator:
Given some linear map f : V ↦ V (where V is a finite-dimensional vector space) generally, we can define the trace of this map by considering the trace of matrix representation of f, that is, choosing a basis for V and
describing f as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis-independent definition for the trace of a linear map.