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Solving an equation for a unitary matrix

  1. Jan 7, 2009 #1
    1. The problem statement, all variables and given/known data

    I have an equation for a unitary matrix [tex]U[/tex],
    [tex]\sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} - \tilde{\epsilon}_k \delta_{qm}} \right)} = 0[/tex]
    I need to solve this equation for [tex]U[/tex]

    2. Relevant equations

    The property of unitarity requires that [tex]U\bar{U} = \hat{I}[/tex]

    3. The attempt at a solution
    If [tex]q \neq m[/tex] then
    [tex]\sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} } \right)} = 0[/tex]
    so that
    [tex]\sum_k \left(\varepsilon_k - \mu\right)\bar{U}_{qk} U_{km} = - \gamma \sum_{kp} \bar{U}_{qk} U_{pm}[/tex]

    If [tex]q = m[/tex] then
    [tex]\sum_k \left(\varepsilon_k - \mu\right) = - \sum_{kp} \left(U_{mk} U_{pm} - \tilde{\epsilon_k}\right)[/tex]

    How do I combine these two results in one equation for [tex]U[/tex]?
  2. jcsd
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