##  [Trace](/trace-1) 

 Definition

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

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

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

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

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

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

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

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

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.