## A 3*3 matrix problem

$A_1,A_2,A_3$ 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 $B_i=u_{i1}*A_1+u_{i2}*A_2+u_{i3}*A_3$?

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I don't see a question here. Do you mean to replace "If there always exists" with "Does there always exist"? – S. Carnahan Oct 26 2010 at 13:44
The typo is corrected. Thanks. – gondolf Oct 27 2010 at 1:40
No, Dennis Serre corrected it and you've un-corrected it. – JBL Oct 27 2010 at 2:19
@JBL. Denis ! My parents were not rich enough to give me two n's. – Denis Serre Nov 2 2010 at 15:56
I have tweaked some of the LaTeX and corrected the English – Yemon Choi Nov 8 2010 at 0:30
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