The matrix completion problem regarding various classes of P0,1- matrices

Abstract

<p>A pattern of positions in an n x n real matrix is said to have pi-completion if every partial pi-matrix which specifies that pattern can be completed to a pi-matrix. An n x n partial matrix A (and the pattern describing the specified entries of A) is associated with a digraph G(V E) where the vertices V of G are 1, 2, 3,...,n, and an arc (i, j) [is in] E if and only if ail is specified in A. A P 0,1-matrix is a matrix in which all principal minors are non-negative and the diagonal is positive. A matrix A is sign symmetric if either aijaji > 0 or a ij = aji = 0 for all entries aij in A. A matrix A is weakly sign symmetric if aijaji > 0 for all entries aij in A. In this dissertation, all digraphs up through order 4 are classified as to whether or not every partial pi-matrix specifying the digraph can be completed to a pi-matrix for pi either of the classes weakly sign symmetric P0,1-matrices, and sign symmetric P 0,1-matrices. Also, all symmetric digraphs of size 5 and 6 are classified as to completion for the class weakly sign symmetric P 0,1-matrices, and all symmetric digraphs of order 4 are classified for the class P0,1-matrices. A number of digraphs of size up through 4 are also classified as to whether or not every partial P0,1-matrix specifying the digraphs can be completed to a P0,1-matrix. A general result is proved that allows the assumption that the non-zero entries associated with up to n - 1 arcs of a subdigraph containing no semicycle are all equal to one in a partial matrix being completed.</p>