[QUE/VS-11001]

Let \(A\) be an operator having eigenvalues 4 and 9 and corresponding
eigenvectors are
\begin{equation*}
  e_1 =\begin{pmatrix} 1 \\ i \end{pmatrix};\qquad
 e_2=\begin{pmatrix}-1\\i\end{pmatrix}\\[-6mm]
\end{equation*} For an arbitrary vector \(f=\begin{pmatrix}\alpha \\ \beta \end{pmatrix}\) find the vectors

1.  \(A f\) 
2. \(\sqrt{A} f\)
3. \(A^2 f\)
4. \(\exp\Big(i\pi A/6 f \Big)\)

Write the matrices for (i) \(\sqrt{A}\), (ii) \(\exp\big(i\pi A/6 f\big)\).