Traceless matrix ($\mathrm{Tr}A=0$)

A traceless matrix is a square matrix whose trace is zero. Such matrices form the kernel of the trace linear functional.

Every commutator of square matrices is traceless by cyclicity of the trace. Conversely, off-diagonal matrix units and differences of diagonal units are themselves commutators and span all traceless matrices. This proves equality with the linear span of commutators, without needing a theorem about representation as one commutator.