= Solution
The <characteristic polynomial> of the <Hermitian matrix> is
$$
\det(A-\lambda I)=\left(\frac32-\lambda\right)^2-\frac{25}{4}=(\lambda-4)(\lambda+1).
$$
Solving the two <eigenvector> equations and using the <Hermitian inner product> to normalize gives
$$
\boxed{\lambda_1=4,\quad v_1=\frac1{\sqrt2}\binom{i}{1};\qquad
\lambda_2=-1,\quad v_2=\frac1{\sqrt2}\binom{-i}{1}.}
$$
Indeed $v_1^\dagger v_2=0$ and both <norms> equal one. Multiplication of either <eigenvector> by any unit <phase factor> gives an equally valid normalized choice.
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