Trace of a matrix

We define the trace of a square matrix as the sum of the elements on its main diagonal.

$$trA=\sum _{i=1}^n{A}_{ii}$$

Some properties of matrix derivatives:

1. $trAB=trBA$
2. $trABC=trCAB=trBCA$
3. $f(A)=trBA\in \mathbb{R}$ where ${\nabla }_{A}trAB=B^T$
4. $trA=tr{A}^T$
5. If $a\in \mathbb{R}\to \text{tr }a=a$
6. ${\nabla }_{a}trAB{A}^{T}C=CAB+C^{T}A{B}^T$

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tags: linear-algebra