The doubly negative matrix completion problem

• Author(s): Araújo, C. Mendes ; Torregrosa, Juan R. ; Urbano, Ana M.
• Date: 2005
• Persistent ID: http://hdl.handle.net/1822/1502
• Origin: RepositóriUM - Universidade do Minho
• Subject(s): Partial matrix; Matrix completion problems; DN-matrix; Undirected graphs
• Document Type: Article
• Language: English

Description

An $n\times n$ matrix over the field of real numbers is a doubly negative matrix if it is symmetric, negative definite and entry-wise negative. In this paper, we are interested in the doubly negative matrix completion problem, that is when does a partial matrix have a doubly negative matrix completion. In general, we cannot guarantee the existence of such a completion. In this paper, we prove that every partial doubly negative matrix whose associated graph is a $p$-chordal graph $G$ has a doubly negative matrix completion if and only if $p=1$. Furthermore, the question of completability of partial doubly negative matrices whose associated graphs are cycles is addressed.