Matrix completion problems for pairs of related classes of matrices

Abstract

For a class X of real matrices, a list of positions in an n× n matrix (a pattern) is said to have X-completion if every partial X-matrix that specifies exactly these positions can be completed to an X-matrix. If X and X 0 are classes that satisfy the conditions any partial X-matrix is a partial X 0-matrix, for any X 0-matrix A and ε>0, A+ εI is a X-matrix, and for any partial X-matrix A, there exists δ>0 such that A−δ I ̃ is a partial X-matrix (where Ĩ is the partial identity matrix specifying the same pattern as A) then any pattern that has X 0-completion must also have X-completion. However, there are usually patterns that have X-completion that fail to have X 0-completion. This result applies to many pairs of subclasses of P- and P 0-matrices defined by the same restriction on entries, including the classes P/ P 0-matrices, (weakly) sign-symmetric P/ P 0-matrices, and non-negative P/ P 0-matrices. It also applies to other related pairs of subclasses of P 0-matrices, such as the pairs classes of P/ P 0,1-matrices, (weakly) sign-symmetric P/ P 0,1-matrices and non-negative P/ P 0,1-matrices. Furthermore, any pattern that has (weakly sign-symmetric, sign-symmetric, non-negative) P 0-completion must also have (weakly sign-symmetric, sign-symmetric, non-negative) P 0,1-completion, although these pairs of classes do not satisfy condition (3). Similarly, the class of inverse M-matrices and its topological closure do not satisfy condition (3), but the conclusion remains true, and the matrix completion problem for the topological closure of the class of inverse M-matrices is solved for patterns containing the diagonal.