If $A$ is any $m \times n$ entrywise non negative matrix, is it true that, if all initial minors of $A$ are nonnegative then $A$ is totally nonnegative (TN)?
I know the analogous result is true for TP, i.e. if all initial minors of $A$ are positive then $A$ is totally positive (TP). I need help in the TN case. In case it's not, is it true for every Hankel matrix? Any proof, tip and/or suggestions or a counterexample will be really appreciated. Thanks in advance.
This isn't true. Consider the Hankel matrix $\pmatrix{0&0&0\\ 0&0&1\\ 0&1&0}$ for instance.