ELA, Volume 7, pp. 1-20, February 2000, abstract.
The General Totally Positive Matrix Completion Problem
with Few Unspecified Entries
Shaun M. Fallat, Charles R. Johnson, and Ronald L. Smith
For m-by-n partial totally positive matrices with exactly one
unspecified entry, the set of positions for that entry that
guarantee completability to a totally positive matrix are
characterized. They are the positions (i, j), i + j less than
or equal to 4 and the positions (i, j), i + j greater than or
equal to m + n - 2. In each case, the set of completing entries
is an open (and infinite in case i=j=1 or i=m, j=n) interval.
In the process some new structural results about totally positive
matrices are developed. In addition, the pairs of positions that
guarantee completability in partial totally positive matrices with
two unspecified entries are characterized in low dimensions.