(non-) Positive Matrix Completion Of The Study, Under The Non-monotonic Calibration Diagram

This paper is devoted to apply the methods of graph theory,combinatorial mathematics in combination with special matrices and linear algebra to discuss the completion problem of totally positive matrix and totally non-positive matrix, whose associated graph is not monotonically labeled. The main contents are as follows:1. The background of this paper and some current state of matrix completion theory and the research method, and our purposes of this paper are presented.2. Through discussing a partial totally positive matrix of size 4×4, whose associ -ated graph is a cycle that is not monotonically labeled, we obtain the "TP-condition" , then we can get the totally positive matrix completion. And we extend the result to an n×n partial totally positive matrix completion problem, whose associated graph is a non-monotonically cycle.3. We discuss the totally non-positive completion of a partial totally non-positive matrices, whose associated graphs are non-monotonically labeled graphs, including non-monotonically labeled paths, non-monotonically labeled 1-chordal graphs, non-monotonically labeled 2-chordal graphs and non-monotonically labeled cycles, and we obtain the "TNP-condition", then we can get the totally non-positive matrix completion.