trace

The trace of a square matrix is the simplest summary you can take: just add up the entries on the main diagonal (top-left to bottom-right). For A = [[2,1],[0,3]], the trace is 2 + 3 = 5. That is the entire recipe.

What makes this humble sum interesting is a beautiful fact: the trace always equals the sum of the eigenvalues. So without finding a single eigenvector you already know what the stretch factors add up to. It pairs naturally with the determinant, which equals the product of the eigenvalues.

The trace is also an invariant: change to any other basis and the diagonal entries shift around, yet their sum stays the same. Because it does not depend on the coordinate system you happen to use, the trace captures something real about the transformation itself.