Trace - Knowlpedia

A trace is the scalar associated to an $n\times n$ matrix $A=(a_{ij})$ over a field $\mathbb{F}$ defined by

\operatorname{tr}(A)=\sum_{i=1}^n a_{ii}.

For a linear operator $T:V\to V$ on a finite-dimensional vector space, the trace $\operatorname{tr}(T)$ is defined as $\operatorname{tr}(A)$ for any matrix $A$ representing $T$ in a basis; this is independent of the basis.

Remarks

Trace is often paired with the determinant in matrix identities and appears in coefficients of the characteristic polynomial. When the characteristic polynomial splits, the trace equals the sum of the eigenvalues counted with algebraic multiplicity.

Examples

If $A$ is diagonal with diagonal entries $d_1,\dots,d_n$ , then $\operatorname{tr}(A)=d_1+\cdots+d_n$ .
The identity matrix $I_n$ satisfies $\operatorname{tr}(I_n)=n\cdot 1_{\mathbb F}$ .
Any nilpotent matrix (some power equals $0$ ) has trace $0$ .