The symmetric N-matrix completion problem

• Author(s): Araújo, C. Mendes ; Torregrosa, Juan R. ; Urbano, Ana M.
• Date: 2005
• Persistent ID: http://hdl.handle.net/1822/2873
• Origin: RepositóriUM - Universidade do Minho
• Subject(s): Partial matrix; Matrix completion problems; N-matrix; Undirected graphs
• Document Type: Article
• Language: English

Description

An $n\times n$ matrix is called an $N$-matrix if all its principal minors are negative. In this paper, we are interested in the symmetric $N$-matrix completion problem, that is, when a partial symmetric $N$-matrix has a symmetric $N$-matrix completion. Here, we prove that a partial symmetric $N$-matrix has a symmetric $N$-matrix completion if the graph of its specified entries is chordal. Furthermore, if this graph is not chordal, then examples exist without symmetric $N$-matrix completions. Necessary and sufficient conditions for the existence of a symmetric $N$-matrix completion of a partial symmetric $N$-matrix whose associated graph is a cycle are given.