## 每週問題 January 26, 2015

A matrix satisfying $A^2=I$ is said to be an involutory matrix, and a matrix $B$ satisfying $B^2=B$ is said to be an idempotent matrix. Show that there is a one-to-one correspondence between the set of $n\times n$ involutory matrices and the set of $n\times n$ idempotent matrices.

$A^2=(I-2B)^2=I-4B+4B^2=I$

$A^2=I$，則

$\displaystyle B^2=\left(\frac{I-A}{2}\right)^2=\frac{I-2A+A^2}{4}=\frac{I-A}{2}=B$