Exercise 6.8.23

Since each permutation could be decomposed into several $2$-cycle, interchanging two elements, we may just prove the statement when the permutation is $2$-cycle. Let $A$ be a diagonal matrix and $B$ be the diagonal matrix obtained from $A$ by interchanging the $\mathit{ii}$-entry and the $\mathit{jj}$-entry. Take $E$ to be the elementary matrix interchanging the $i$-th and the $j$-th row. Then we have $E$ is symmetric and $\mathit{EAE}=B$.