Trace
The trace is a number that embeds some information about a matrix, much like the determinant.
Contents
Description
The trace is the sum of the main diagonal of a matrix. For a matrix A whose cells are referenced as aij for all rows i and columns j, the trace is Σi aii.
The trace is also the sum of eigenvalues: Σi λi.
The trace of A is notated as tr(A).
Properties
Only square matrices have a trace.
Transposition does not change the trace: tr(A) = tr(AT).
Traces are linear:
tr(cA) = c tr(A)
tr(A + B) = tr(A) + tr(B)
Given two square matrices of the same size, or two matrices of size m x n and n x m, then tr(AB) = tr(BA).