先證明 \(tr(AB)=tr(BA)\)，
不失一般性假設 \(A\) 是 \(n\times m\) 階矩陣，\(B\) 是 \(m\times n\) 階矩陣，則
\(\displaystyle tr(AB) = \sum_{i=1}^n \left(AB\right)_{ii}=\sum_{i=1}^n \sum_{j=1}^m A_{ij} B_{ji} = \sum_{j=1}^m \sum_{i=1}^n A_{ji} B_{ij} = \sum_{j=1}^m\left(BA\right)_{jj} = tr(BA).\)
利用 \(tr(AB)=tr(BA)\) 可得
\(tr\left(\left(ABC\right)D\right)=tr\left(D\left(ABC\right)\right),\)
\(tr\left(\left(AB\right)\left(CD\right)\right)=tr\left(\left(CD\right)\left(AB\right)\right),\)
\(tr\left(A\left(BCD\right)\right)=tr\left(\left(BCD\right)A\right).\)
