tracefree

Tracefree (traceless) describes a matrix or tensor whose trace, the sum of its diagonal entries, is zero. For a square matrix A in an n-by-n space, the trace is tr(A) = sum of A's diagonal entries. A matrix is trace-free if tr(A) = 0.

Equivalently, any square matrix M can be decomposed into a scalar part and a trace-free part: M = (tr(M)/n) I + M0, where M0 = M − (tr(M)/n) I and tr(M0) = 0. The collection of all trace-free matrices forms the Lie algebra sl(n, F) of the special linear group, with field F typically R or C. The trace is a linear functional, and tr([A,B]) = 0 for any matrices A and B, illustrating how trace-free subspaces interact with commutators.

Trace-free tensors generalize the idea to higher rank. For a second-order tensor T on an n-dimensional space, the trace is T^i_i; its trace-free part is T0 = T − (1/n) tr(T) I, or, in a metric context, T0 = T − (tr_g T)/n g. In physics, traceless tensors and fields are common in gauge theories and in conformal settings, where the trace part corresponds to scaling.

In differential geometry and general relativity, the fully trace-free component of the curvature is captured by the Weyl tensor, which encodes conformal, scale-invariant aspects of curvature. The concept of trace-free components appears across linear algebra, representation theory, and theoretical physics, serving as a natural way to separate scale or scalar parts from more geometric information.

See also: trace, traceless, Weyl tensor, sl(n).