Let $$ \text{U}(\theta;i,j)= \begin{pmatrix} 1 \\ & \ddots \\ && 1 \\ &&&\cos\theta&&&&-\sin\theta \\ &&&&1 \\ &&&&&\ddots \\ &&&&&&1 \\ &&&\sin\theta&&&&\cos\theta \\ &&&&&&&&1 \\ &&&&&&&&&\ddots \\ &&&&&&&&&&1 \end{pmatrix} $$ the rotation matrix on coordinates i,j plane.
How to explain why only rows and columns of indexes i and j are changed in a unitary equivalence like : $$ A = U(\theta;i,j)BU^*(\theta;i,j),\hspace{16pt}A,B \in M_n $$ It seem pretty obvious but I can't put words on it.
A useful tool here is the following fact if $e_1,\dots,e_n \in \Bbb R^n$ denote the standard basis vectors, then the $i,j$ entry of $A$ is given by $A_{ij} = e_i^*A e_j$. With that, note that for any $k \notin \{i,j\}$, the matrix $U = U(\theta; i,j)$ satisfies $U^*e_k = e_k$. It follows that for any $p,q$ (both not equal to $i$ or $j$), we have $$ A_{ij} = e_i^T[UBU^*]e_j = [U^*e_i]^* B [U^*e_j] = e_i^* B e_j = B_{ij}. $$