For a square matrix or endomorphism, the trace is the sum of diagonal entries in any matrix representation; equivalently the sum of eigenvalues counted with algebraic multiplicity and an invariant under similarity.

Principle

Linearity and cyclicity are central: trace(A+B)=trace(A)+trace(B), trace(cA)=c·trace(A), and trace(AB)=trace(BA) for appropriately sized matrices; the coefficient of λ^{n−1} in the characteristic polynomial is −trace(A).

Demonstration

Example: A = [[1,2],[3,4]] has trace 1+4=5. Its eigenvalues sum to 5, and for any invertible S, trace(S^{-1}AS)=trace(A), so trace is similarity-invariant.

Misapplication

Using trace=0 to conclude nilpotency or singularity: a zero trace does not imply a matrix is nilpotent or noninvertible (trace gives only spectral sum information), or using trace to deduce eigenvalues individually.

Consequence

Trace provides a simple, computable invariant that controls coefficients of characteristic polynomial, appears in linearization of determinant, and serves as the character in representation theory; it is additive under direct sums and stable under conjugation.

Reversal

Rather than summarizing a matrix by its trace (a single scalar), analyze the full spectrum or Jordan form to recover detailed spectral structure lost by summation.

Boundary

Defined for square matrices over fields and for trace-class operators in infinite dimensions; for general infinite-dimensional operators trace may be undefined and depend on choice of basis or topology.

Semantic Tension

Tension between trace and determinant: trace aggregates eigenvalues additively while determinant multiplies them; also tension between trace as basis-dependent diagonal sum versus invariant under similarity—apparent paradox resolved by similarity invariance of trace.

Synthesis

Trace is the similarity-invariant scalar giving the sum of diagonal entries or equivalently the sum of eigenvalues; linearity and cyclicity govern its algebraic uses and limitations.