Generalizing the spectral radius of a unistochastic matrix

Question

Consider a square matrix $A$, and from it construct $B$ whose entries are the squared magnitudes of those in $A$. What can we say about the spectral radius of $B$? I know that for a unitary matrix $A$, $B$ is unistochastic so its spectral radius is 1, but I'm interested in the general case for arbitrary $A$.

Also, relatedly, for general $A$, what can be said of the column 1-norms of $B$? And what about the the column 1-norms of products of such $B$-type matrices?

Answer 1

Well, $B$ is the Hadamard product of $A$ with its conjugate, so if $A$ is nonnegative or psd, its radius is bounded from above by the square of the radius of $A$.

For more information see this paper.