Trace

square matrix $\mathbf{A}\in {\mathbb{R}}^{n\times n}$ 의 trace 는 주어진 정사각 행렬의 대각 원소들의 합을 의미한다.

$$\mathrm{tr}(\mathbf{A}):=\sum _{i=1}^n{a}_{ii}$$

B) Properties

• $\mathrm{tr}(\mathbf{A}+\mathbf{B})=\mathrm{tr}(\mathbf{A})+\mathrm{tr}(\mathbf{B})\text{for}\mathbf{A},\mathbf{B}\in {\mathbb{R}}^{n\times n}$
• $\operatorname{tr}(\alpha\boldsymbol{A})=\alpha\operatorname{tr}(\boldsymbol{A}),\alpha\in\mathbb{R}\text{for}\boldsymbol{A}\in\mathbb{R}^{n\times n}$$
• $\mathrm{tr}\left({\mathbf{I}}_n\right)=n$

• \operatorname{tr}(\boldsymbol{A}\boldsymbol{B})=\operatorname{tr}(\boldsymbol{B}\boldsymbol{A})\text{for}\boldsymbol{A}\in\mathbb{R}^{n\times k},\boldsymbol{B}\in\mathbb{R}^{k\times n}

\operatorname{tr}\left(\boldsymbol{x}\boldsymbol{y}^{\top}\right)=\operatorname{tr}\left(\boldsymbol{y}^{\top}\boldsymbol{x}\right)=\boldsymbol{y}^{\top}\boldsymbol{x}\in\mathbb{R} $$, for $\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{n}$ # Related # References