# 问题：

Show that the identity operator on a vector space $V$ has a matrix representation which is one along the diagonal and zero everywhere else, if the matrix representation is taken with respect to the same input and output bases. This matrix is known as the identity matrix.

## 解答

$\begin{array}{c} A\left|v_{1}\right\rangle=1\left|v_{1}\right\rangle+0\left|v_{2}\right\rangle+\cdots+0\left|v_{n}\right\rangle \\ A\left|v_{2}\right\rangle=0\left|v_{1}\right\rangle+1\left|v_{2}\right\rangle+\cdots+0\left|v_{n}\right\rangle \\ \vdots \\ A\left|v_{n}\right\rangle=0|v_{1}\rangle+0\left|v_{2}\right\rangle+\cdots+1\left|v_{n}\right\rangle \end{array}$.

$\left(\begin{array}{cccc} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & 1 \end{array}\right)$.

#### 参考

[1]www.qtumist.com