A gate for a Quantum Computer

C-U=∣0⟩⟨0∣⊗I+∣1⟩⟨1∣⊗UC\text{-}U = \ket{0}\bra{0} \otimes I + \ket{1}\bra{1} \otimes U

i.e.,

Let

U=(u00u01u10u11)U = \begin{pmatrix} u_{00} & u_{01} \\ u_{10} & u_{11} \end{pmatrix}

then

C-U=(1000010000u00u0100u10u11)C\text{-}U = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & u_{00} & u_{01} \\ 0 & 0 & u_{10} & u_{11} \end{pmatrix}