Journal or Book Title
Linear Algebra and its Applications
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 X0-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/P0-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.
This manuscript version is made available under the CCBY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Hogben, Leslie, "Matrix completion problems for pairs of related classes of matrices" (2003). Mathematics Publications. 74.