##  [Trace](/trace-3) 

 Definition

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

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

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

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

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

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

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

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

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.