# 问题：

Suppose $A’$and $A’’$ are matrix representations of an operator $A$ on a vector space $V$ with respect to two different orthonormal bases, $\left|v_{i}\right\rangle$ and $\left|w_{i}\right\rangle$. Then the elements of $A’$ and $A’’$ are$A_{i j}^{\prime}=\left\langle v_{i}|A| v_{j}\right\rangle$and$A_{i j}^{\prime\prime}=\left\langle w_{i}|A| w_{j}\right\rangle$ . Characterize the relationship between $A’$ and $A’’$.

## 解答

$$\left\langle v_{i}|A| v_{j}\right\rangle=\sum_{kl }\left\langle v_{i} \mid w_{k}\right\rangle\left\langle w_{k}|A| w_{l}\right\rangle\left\langle w_{l} \mid v_{j}\right\rangle .$$

#### 参考

[1]www.qtumist.com