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.