Let $D(\lambda_1, \cdots, \lambda_n)$ denote the $n\times n$ diagonal matrix whose $(i,i)$ entry is $\lambda_i$.
For any permutation, $\sigma \in S_n$, I want to show that $D(\lambda_{\sigma(1)}, \cdots, \lambda_{\sigma(n)} )$ is similar to $D(\lambda_1, \cdots, \lambda_n)$ where $S_n$ means the symmetric group on $n$ letters.
Intuitively, since $D(\lambda_{\sigma(1)}, \cdots, \lambda_{\sigma(n)} )$ is nothing but shuffling the order of diagonal matrix, I know it is diagonal but have no idea to show this is similar to $D(\lambda_1, \cdots, \lambda_n)$