[선형대수] Trace

Trace는 행렬의 대각 원소를 전부 더한 값을 의미하며, 정사각 행렬에서만 정의된다.

\[\text{tr}(A)=\sum_{i=1}^na_{ii} \tag{1}\]

Trace의 성질

1. $\text{tr}(A+B)=\text{tr}(A)+\text{tr}(B)$
2. $\text{tr}(\lambda A)=\lambda\text{tr}(A)$
3. $\text{tr}(I_n)=n$
4. $A\in\mathbb{R}^{\color{red}n\times \color{blue}k},B\in\mathbb{R}^{\color{blue}k\times \color{red}n}\implies\text{tr}(AB)=\text{tr}(BA)$
5. $A\in\mathbb{R}^{\color{red}a\times \color{blue}k},B\in\mathbb{R}^{\color{blue}k\times \color{green}l},C\in\mathbb{R}^{\color{green}l\times \color{red}n}\implies\text{tr}(ABC)=\text{tr}(CAB)=\text{tr}(BCA)$
6. $\text{tr}(\mathbf{x}^\top\mathbf{y})=\mathbf{x}^\top\mathbf{y}\in\mathbb{R}$
7. $\text{tr}(A)=\text{tr}(S^{-1}AS)$
8. $\text{det}(I+\Delta)=1+\text{tr}(\Delta)$