$A_1,A_2,A_3$ are $3\times3$ matrices with $$tr(A_1^{\dagger}A_2)=0,$$ $$tr(A_2^{\dagger}A_3)=0,$$ $$tr(A_3^{\dagger}A_1)=0,$$ and $tr(A_1^{\dagger}A_1)=tr(A_2^{\dagger}A_2)=tr(A_3^{\dagger}A_3)$, where $A^{\dagger}$ denotes the conjugate transposed matrix of $A$.
Does there always exist a $3\times3$ complex unitary $U=[u_{ij}]_{3*3}$ such that one can find a unitary $V$ which sets the diagonal elements of $V^{\dagger}B_i^{\dagger}B_jV$ to 0 for all $i\neq j$, where $B_i=u_{i1}*A_1+u_{i2}*A_2+u_{i3}*A_3$?
