THE $R$-MATRIX COMPLETION PROBLEM

Matrix completion problem (MCP) is a very well-established process of rebuilding a matrix's unknown elements. A $m\times m$ matrix $B$ is a $R$-matrix if for every $r = 1,2,\ldots, m$, the sum of all $r \times r$ principal minors of $B$ is negative. A digraph $D$ possesses $R$-completion if it is possible to complete any partial $R$-matrix that defines $D$ to a $R$-matrix. In this article we have examined the $R$-matrix completion problem. Here some necessary as well as some sufficient conditions for a digraph to have the $R$-completion are discussed. In addition, the digraphs of order up to four that possesses $R$-completion have been categorized. Finally, a comparative discussion between $R$-matrix completion and $N$-matrix completion is provided.