The trace of a square linear endomorphism or matrix is the sum of the diagonal entries of any representing matrix; equivalently it equals the sum of the eigenvalues counted with algebraic multiplicity and is invariant under similarity.

Principle

Trace is a linear, similarity-invariant functional: tr(A + B) = tr(A) + tr(B), tr(cA) = c tr(A), and tr(S−1AS) = tr(A). In characteristic polynomial terms, the trace is (up to sign) the coefficient of λ^{n−1}.

Demonstration

For A = [[1,2],[0,3]] the diagonal entries sum to 1 + 3 = 4, and the eigenvalues are 1 and 3 so their sum is 4; tr(A) = 4 is invariant under change of basis.

Misapplication

Assuming the trace uniquely determines the spectrum or individual eigenvalues; treating trace as dependent on a particular matrix representation rather than recognizing its basis invariance; or misusing trace identities outside their valid algebraic contexts.

Consequence

Trace provides a simple spectral invariant (sum of eigenvalues), is additive over block diagonal decompositions, and satisfies tr([A,B]) = 0 for commutators [A,B] = AB − BA, which has consequences in representation theory and physics.

Reversal

Contrast with determinant, which multiplicatively aggregates eigenvalues (product) and is zero/nonzero to indicate singularity; trace is an additive spectral summary rather than multiplicative.

Boundary

Well-defined for finite-dimensional endomorphisms; for operators on infinite-dimensional spaces the trace may be undefined except for trace-class operators. Trace depends on the field (characteristic issues) and on working with square endomorphisms.

Semantic Tension

Tension with determinant and full characteristic polynomial: trace gives only the first symmetric sum of eigenvalues, so different spectra can share the same trace; tension with matrix-entry intuition arises because similarity changes diagonal entries but preserves trace.

Synthesis

Trace is the linear spectral invariant equal to the sum of diagonal entries or eigenvalues (with algebraic multiplicity), linear and similarity-invariant, useful for quick spectral checks and identities but insufficient to recover the full spectrum.