Traceless

Traceless describes an object whose trace is zero. In linear algebra, a square matrix A is traceless if the trace, the sum of its diagonal entries, equals zero. The trace is invariant under similarity transformations, so tracelessness is a basis-independent property.

A common way to obtain a traceless matrix from any A is to subtract (trace(A)/n) times the identity matrix I, yielding B = A − (trace(A)/n) I. Then tr(B) = 0. The concept extends to tensors, where a trace is formed by contracting indices with a metric or the identity; a traceless tensor has that contraction equal to zero.

In algebra, the set of all n×n traceless matrices over a field forms the Lie algebra sl(n). This algebra plays a central role in the study of the special linear group SL(n), the group of determinant-1 invertible matrices. In physics, traceless matrices often appear as generators of Lie groups; for example, the generators of SU(n) are represented by traceless Hermitian matrices (up to a factor of i), with the Gell-Mann matrices for SU(3) being a notable set.

Traceless tensors also arise in physical contexts. The energy-momentum tensor may be traceless in conformal or scale-invariant theories in four dimensions; the Weyl part of the curvature tensor in general relativity is traceless. Tracelessness typically removes a trace degree of freedom, enforcing a constraint that highlights deviations from uniform scaling or volume effects.

Summary: Traceless denotes zero trace, a property with broad implications in linear algebra, Lie theory, and physics, often achieved by removing the trace component through subtraction of a suitable multiple of the identity.