A Note on Matrix Completion Problems

Abstract

Matrix completion problems are an important subclass of problems in matrix theory. An important question in matrix completion problems was posed by Oliveira in 1975, where the author proposed the description of the characteristic polynomial of a partitioned matrix of the form A = [Ai,j], i, j ∈ {1,2} (whose entries are in a field and A1,1, A2,2 are square submatrices), when some of the blocks Ai,j are prescribed and the others are unknown. The analysis of this problem gave rise to several subproblems, according to the location of the prescribed submatrices. Many authors have considered this list of subproblems. In this note we provide a new proof of a result obtained by Oliveira inserted in this class of subproblems.

The Many Faces of Degeneracy in Conic Optimization

“…The work using chordal graphs for PSD completions was done in [61] and extended the special case of banded structure in [48]. Surveys for matrix completion are given in e.g., [70,68,4,63,32,33,71,69]. A survey specifically related to chordality is given in [139].…”