Are diagonal elements of a matrix dominated by its singular values?

Question

Let $A$ be an $n \times n$ real matrix with $\det A \ge 0$. Denote by $\sigma_i$ the singular values of $A$.

Does there exist a permutation $\alpha \in S_n$ such that $A_{ii} \le \sigma_{\alpha(i)}$ for every $1 \le i \le n$?

I am particularly interested in the case where $n=2$.

Answer 1

No, think of $\begin{pmatrix}N+1 & N \\ N & N+1\end{pmatrix}$; one eigenvalue (and singular value) is $1$ but diagonal entries are big.

Answer 2

When the diagonal entries are positive, a necessary condition is that the product of the diagonal entries is at most $\det(A)$.

Take $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $a, d > 0$, $bc < 0$. Then $ad < \det(A)$.