Is there a constant $c$ such that $\sigma_1(A)\leq c\rho(A)$ for all $n\times n$ matrices $A$?

Question

Let $A$ be a $n\times n$ matrix. Denote by $\sigma_1(A)$ the largest singular value of $A$, and $\rho(A):=\max_\mu\{{|\lambda_\mu|}\}$ the radius of the eigenvalues $\lambda_\mu$ of $A$.

As is well-known, there is \begin{eqnarray} \rho(A)\leq\sigma_1(A). \end{eqnarray}

My question is the converse:

Is there a constant $c$ such that \begin{eqnarray} \sigma_1(A)\leq c\rho(A), \end{eqnarray} for all $n\times n$ matrices $A$?

Answer 1

No. Consider for instance $$A=\begin{bmatrix}0&1\\0&0\end{bmatrix}.$$ The answer is yes, with $c=1$, when $A$ is selfadjoint.