= Solution
{c}
The <partial trace> removes the other orthonormal factor in the <Schmidt decomposition>:
$$
\boxed{\rho_A=\sum_kp_k|a_k\rangle\langle a_k|,\qquad
\rho_B=\sum_kp_k|b_k\rangle\langle b_k|.}
$$
For instance, tracing $|a_kb_k\rangle\langle a_\ell b_\ell|$ over $B$ gives $\langle b_\ell|b_k\rangle|a_k\rangle\langle a_\ell|$, which vanishes unless $k=\ell$. Consequently the nonzero <eigenvalues> of the two <reduced density operators> agree, with multiplicity. If the subsystem dimensions differ, the orthonormal Schmidt families can be completed to full bases and the extra eigenvalues are zero.
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