Completion Problems for Positive Matrices with Minimal Rank

The real positive definite completion problem for a 4-cycle

Bounds for minimum semidefinite rank from superpositions and cutsets

Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property

Unitary matrix digraphs and minimum semidefinite rank

Matrices attaining the minimum semidefinite rank of a chordal graph

The copositive completion problem: Unspecified diagonal entries

formula omitted] completions on partial matrices

Circulant partial Hadamard matrices are useful in many scientific applications, yet the existence of such matrices is not known in general. Some Hadamard matrices of given orders are found via complete enumeration in the literature, but the searches are too computationally intensive when the orders are large. This paper introduces a search method for circulant partial Hadamard matrices by using natural heuristic algorithm. Slightly deviated from the swarm intelligence based algorithm, this method successfully generates a class of circulant partial Hadamard matrices efficiently. A table of circulant partial Hadamard matrices results are given at the end of this paper.

Given an undirected graph G=(V,E) with node set V=[1,n], a subset $S\subseteq V$, and a rational vector $a\in {\rm\bf Q}^{S\cup E}$, the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n × n positive semidefinite matrix X=(xij) satisfying xii=ai ($i\in S$) and xij=aij ($ij\in E$). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in, the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4.

Rectangular submatrices of inverse [formula omitted]-matrices and the decomposition of a positive matrix as a sum

Minimum rank problems

Special Issue on Nonnegative matrices, M-matrices and their generalizations

On minimum rank and zero forcing sets of a graph

Existence and construction of nonnegative matrices with complex spectrum

Decompositions of minimum rank matrices