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 X0 are classes that satisfy the conditions any partial X-matrix is a partial X0-matrix, for any X0-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-δĨ is a partial X-matrix (where Ĩ is the partial identity matrix specifying the same pattern as A)then any pattern that has X0-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 P0-matrices defined by the same restriction on entries, including the classes P/P0-matrices, (weakly) sign-symmetric P/P 0-matrices, and non-negative P/P0-matrices. It also applies to other related pairs of subclasses of P0-matrices, such as the pairs classes of P/P0,1-matrices, (weakly) sign-symmetric P/P0,1-matrices and non-negative P/P0,1-matrices. Furthermore, any pattern that has (weakly sign-symmetric, sign-symmetric, non-negative) P0-completion must also have (weakly sign-symmetric, sign-symmetric, non-negative) P0,1-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. © 2003 Elsevier Inc. All rights reserved.
Hogben, L. (2003). Matrix completion problems for pairs of related classes of matrices. In Linear Algebra and Its Applications (Vol. 373, pp. 13–29). https://doi.org/10.1016/S0024-3795(02)00531-1