# Characterizing invertible nonnegative matrices with bounded sums

Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample showing that there does not exist such a bound.

I am actually more interested in the following (presumably much harder) question:

Characterize the class of $n \times n$, elementwise nonnegative, semidefinite matrices with entries in $[0,1]$, whose pseudoinverse satisfies $e^TA^{\dagger}e \le n$?

Here $A^\dagger$ denotes the Moore-Penrose pseudoinverse and $e$ denotes the vector of all ones.

In particular, I am seeking equivalent conditions so that given an $A$, one can decide whether it belongs to the said class or not. For example, $A=I_n$ belongs to this class, as do the matrices described in this question (with $\alpha=1/n$).

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