How to show that $(Re\lambda_1,\dotsb,Re\lambda_n)^T$ is majorized by $(\lambda_1(H),\dotsb,\lambda_n(H))^T$?
Solution 1:
$A= H + iK = \frac{1}{2}\big(A+A^*\big)+\frac{1}{2}\big(A-A^*\big)$.
via congruence (i.e. using $Q^*AQ$ where $A=QRQ^*$ in Schur Triangularization) we can assume WLOG that $A$ is upper triangular.
Then
$\max_{1\leq i_1<\dotsb <i_k\leq n }\sum_{j=1}^k Re\lambda_{i_j}= \max_{1\leq i_1<\dotsb <i_k\leq n }\sum_{j=1}^k h_{i_j}$
i.e. maximal sum of k elements of the diagonal of $\frac{1}{2}\big(A+A^*\big)$
and this is well known to be majorized by the eigenvalues of $H$. Reference e.g. one of the many proofs here: Maximize $\mathrm{tr}(Q^TCQ)$ subject to $Q^TQ=I$