For a square matrix or an endomorphism of a finite-dimensional vector space, the trace is the sum of the diagonal entries (with respect to any matrix representation) and equals the sum of the eigenvalues counted with algebraic multiplicity. It is invariant under similarity transformations.

Principle

The trace is a linear, similarity-invariant scalar summary of an endomorphism that is equal to the contraction of a matrix with the identity and equals the sum of eigenvalues in algebraically well-behaved settings.

Demonstration

For A = [[2,1],[0,3]] the trace tr(A)=2+3=5; if A has eigenvalues 2 and 3 (with algebraic multiplicity) their sum also equals 5.

Misapplication

Treating the diagonal-sum of a non-square or infinite matrix as a trace without checking the linear-algebraic context, or assuming the eigenvalue-sum identity holds over arbitrary rings or for non-trace-class operators.

Consequence

Correct use yields a coordinate-independent invariant that appears in characteristic polynomials, in the derivative of determinant at the identity, and in physics as the expectation of linear observables; it can simplify classification of operators up to similarity.

Reversal

The inversion contrasts with determinant: while trace sums eigenvalues, determinant multiplies them; setting trace to zero does not imply nilpotence unless further conditions hold.

Boundary

Defined straightforwardly for finite square matrices and trace-class operators; for infinite-dimensional operators a trace exists only for trace-class operators; over arbitrary rings eigenvalue statements may fail or be meaningless.

Semantic Tension

Competes with 'partial trace' (a reduction operation in tensor-product spaces) and with coordinate-dependent diagonal-sum — the latter may change under change of basis if similarity invariance is ignored.

Synthesis

Trace is the basis-independent scalar obtained by summing matrix diagonal entries in finite dimensions; it organizes linear maps by a linear invariant equal to the eigenvalue sum in appropriate algebraic contexts, but must be restricted or generalized carefully outside finite-dimensional vector spaces.