Interchange of diagonal elements with unitary transformation

If $(e_1,e_2,e_3)$ is the basis in which the first matrix $A$ is written, you have

$$begin{cases} A(e_2) &= b e_2\ A(e_3) & = c e_3 end{cases}$$

So if you exchange $e_2$ and $e_3$, you'll have the desired result.

Therefore the unitary transformation you're looking for is

$$begin{pmatrix} 1 & 0 & 0\ 0 & 0 & 1\ 0 & 1 & 0 end{pmatrix}$$

Just multiply with the permutation matrix $$P=begin{bmatrix}1&0&0\0&0&1\0&1&0end{bmatrix}$$ as follows: $A_2=P^top A_1 P$. Permutation matrix is unitary and orthogonal: $$bar P^top P=P^top P=I.$$