# 问题：

If $A$ and $B$ are two linear operators, show that $tr(A + B) = tr(A) + tr(B)$ (2.63)

and if $z$ is an arbitrary complex number show that $tr(zA) = ztr(A)$.

## 解答

$\operatorname{tr}(A+B)=\sum_{i}\left(A_{i i}+B_{i i}\right)=\sum_{i} A_{i i}+\sum_{i} B_{i i}=\operatorname{tr}(A)+\operatorname{tr}(B)$；

$\operatorname{tr}(z A)=\sum_{j} z A_{i i}=z \sum_{j} A_{i i}=z \operatorname{tr}(A)$.

#### 参考

[1]www.qtumist.com