基础性质
设$r$为标量，则有
\[tr(\pmb{A}+\pmb{B}) = tr(\pmb{A}) + tr(\pmb{B})\] \[tr(r\pmb{A}) = r \cdot tr(\pmb{A})\] \[tr(\pmb{A}) = tr(\pmb{A}^T)\]矩阵乘积的迹
\[tr(\pmb{A}\pmb{B}) = tr(\pmb{B}\pmb{A})\] \[tr(\pmb{A}\pmb{B}\pmb{C}) = tr(\pmb{B}\pmb{C}\pmb{A}) = tr(\pmb{C}\pmb{A}\pmb{B})\]当矩阵$\pmb{A}, \pmb{B}, \pmb{C}$为$n \times n$的对称矩阵时，如下矩阵乘积转换成立：
\[\begin{align} tr(\pmb{A}\pmb{B}\pmb{C}) &= tr(\pmb{B}\pmb{C}\pmb{A}) = tr(\pmb{C}\pmb{A}\pmb{B}) \\ &=tr(\pmb{A}\pmb{C}\pmb{B}) = tr(\pmb{C}\pmb{B}\pmb{A}) = tr(\pmb{B}\pmb{A}\pmb{C}) \end{align}\]