Trace

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

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.

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 $\operatorname{tr}(I_n)=n$.
• Any nilpotent matrix (some power equals $0$) has trace $0$.