Diagonalize (e^{sigma_{1}}), where (sigma_{1}) is the first Pauli matrix (left(begin{array}{ll}0 & 1 1 & 0end{array} ight)).

Diagonalize \(e^{\sigma_{1}}\), where \(\sigma_{1}\) is the first Pauli matrix \(\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)\).