# An issue with unitary matrices

1. Jan 6, 2009

### Threepwood

1. The problem statement, all variables and given/known data

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

2. Relevant equations

The property of unitarity requires that $$U\bar{U} = \hat{I}$$

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

If $$q = m$$ then
$$\sum_k \left(\varepsilon_k - \mu\right) = - \sum_{kp} \left(U_{mk} U_{pm} - \tilde{\varepsilon_k}\right)$$

How do I combine these two results in one equation for $$U$$?