4.8 Trace

Trace of Matrix

The trace of a matrix is the sum of its diagonal entries. Formally, for a general matrix $A\in {\mathbb{R}}^{m\times n}$,
$\mathrm{Tr}(A)={\sum }_{i=1}^{\min \{m,n\}}{A}_{ii}.$

Properties

$\mathrm{tr}(ABC)=\mathrm{tr}(BCA)=\cdots =\mathrm{tr}(CAB)$

Trace is invariant under cyclic permutations.

$\mathrm{tr}(\vec{b}{\vec{a}}^T)={\vec{a}}^T\vec{b}$

The trace (sum of products) of the outer product of two matrices is equal to the dot product of those two matrices.

$|A{|}_F=\sqrt{\mathrm{tr}(A^{T}A)}$

$$\begin{array}{rl}\mathrm{tr}(A^{T}A)& =\sum _{i=1}^r(A^{T}A{)}_{ii}\\ & =\sum _{i=1}^r{A}_{ii}^T{A}_{ii}\\ & =\sum _{i=1}^r{A}_{ii}^2\end{array}$$

for $r=\min \{m,n\}$.

$\mathrm{tr}(A)={\sum }_{i=1}^n{\lambda }_i$