Quantum computation and quantum ... - Google Books

I am doing the problem above.

$\displaystyle A_{ij} = < v_i| T | v_j>$ then use the fact that $\displaystyle I = \sum_m |w_m><w_m|$ and $\displaystyle I = \sum_n |w_n><w_n|$

eventually you get

$\displaystyle A_{ij} = \sum_m \sum_n <v_i|w_m><w_n|v_j>B_{mn}$

how do I invoke the unitary matrix? We know that $\displaystyle U=|w_i><v_i|$