Eigenvalues of the absolute value of the matrix $A$

Question

Can we say that there is a relationship between the eigenvalues of the matrix $A$ and its absolute value as $B$, where $b_{ij}=|a_{ij}|$? Consequently, can we say that there is a relationship between the spectral radius of them?

Answer 1

Let $A\in M_n(\mathbb{C})$. According to Wielandt, if $B$ is irreducible, then $\rho(A)\leq \rho(B)$.

EDIT. In fact, the inequality remains true even if $B$ is reducible.

Indeed, there is a sequence of positive matrices $(B_k)$ s.t. $B\leq B_k$ and $(B_k)$ tends to $B$.

Thus $|A|\leq B_k$ and $B_k>0$ implies that $\rho(A)\leq \rho(B_k)$. The conclusion comes from $\rho(B_k)$ tends to $\rho(B)$.