TITLE: Do singular values dominate eigenvalues?
QUESTION [9 upvotes]: Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that  $|\lambda_1| \ge |\lambda_2| \ge \cdots \ge |\lambda_n|$. Is it true that $s_1 \ge |\lambda_1|,\ s_1+s_2 \ge |\lambda_1| + |\lambda_2|,$ and so on?

REPLY [10 votes]: Let $\lambda(A)$ denote the vector of eigenvalues and $s(A)$ the vector of singular values (arranged in decreasing order). The claim of the question is whether $|\lambda(A)|^{\downarrow} \prec_w s(A)$. The answer to this question is yes (the proof follows using the tensor power trick Yanqi Qui mentioned above).

Theorem (Weyl's Majorant theorem). Let $f: (0,\infty)\to(0,\infty)$ be such that $f(e^t)$ is convex and monotone increasing in $t$. Then
  $$[f(\vert\lambda_1\vert),\ldots, f(|\lambda_n|)] \prec_w [f(s_1),\ldots,f(s_n)].$$

As a corollary, using the function $t \mapsto t^p$ for any $p \ge 0$, we obtain an affirmative answer to the OP's question (actually, by merely using the special case $p=1$). Actually, Weyl's Majorant theorem provides the following log-majorization (which implies the above result):
\begin{equation*}
   \log |\lambda(A)| \prec \log s(A),
\end{equation*}
and this version also answers Ilya's comment, namely $\prod_i^n |\lambda_i| = \prod_i^n s_i$.
Another closely related result is that of absolute values of the Hermitian part of a complex matrix. Let $\Re(A) := (A+A^*)/2$. Then, we have

Theorem (Fan-Hoffman). For every matrix $A$, we have
  \begin{eqnarray*}
   \lambda_j^{\downarrow}(\Re(A)) &\le& s_j(A),\qquad 1\le j \le n\\
   |\lambda(\Re(A))| &\prec_w& s(A).
\end{eqnarray*}

Reference
R. Bhatia, Matrix Analysis (Springer GTM, 169).