= Solution
The normalized circular <eigenvectors> from part (a) define the <orthogonal projection> matrices
$$
P_+=\mathbf u_+\mathbf u_+^\dagger=\frac12\begin{pmatrix}1&i\\-i&1\end{pmatrix},\qquad
P_-=\mathbf u_-\mathbf u_-^\dagger=\frac12\begin{pmatrix}1&-i\\i&1\end{pmatrix}.
$$
Thus the <spectral decomposition> of the <Jones matrix> is
$$
\boxed{J=\lambda_+P_++\lambda_-P_-
=\frac12\begin{pmatrix}
\lambda_++\lambda_-&i(\lambda_+-\lambda_-)\\
-i(\lambda_+-\lambda_-)&\lambda_++\lambda_-
\end{pmatrix}.}
$$
This simply substitutes $a=(\lambda_++\lambda_-)/2$ and $b=(\lambda_+-\lambda_-)/2$ and makes each <eigenmode>'s <wave amplitude> transmission explicit.
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