On completion problems for various classes of P-matrices

Abstract

A P-matrix is a real square matrix having every principal minor positive, and a Fischer matrix is a P-matrix that satisfies Fischer's inequality for all principal submatrices. In this paper, all patterns of positions for n × n matrices, n ≤ 4, are classified as to whether or not every partial Π-matrix can be completed to a Π-matrix for Π any of the classes positive P-, nonnegative P-, or Fischer matrices. Also, all symmetric patterns for 5 × 5 matrices are classified as to completion of partial Fischer matrices, and all but 2 such patterns are classified as positive Por nonnegative P-completion. We also show that any pattern whose digraph contains a minimally chordal symmetric-Hamiltonian induced subdigraph does not have Π-completion for Π any of the classes positive P-, nonnegative P-, Fischer matrices.

Completion of Hankel Partial Contractions of Non-Extremal Type

“…optimization, see, for instance, [19]), electrical engineering (e.g. data transmission, coding and image enhancement, see, for instance, [3]) and geophysics (seismic reconstruction problems, see, for instance, [14]). A Hankel Partial Contraction (HPC) is a square Hankel matrix such that not all of its entries are determined, but in which every well-defined submatrix (completely determined submatrix) is a contraction (in the sense that their operator norms are at most 1).…”

Completion of Hankel partial contractions of extremal type

“…On the other hand, matrix completion problems arise naturally in (i) probability and statistics (e.g. entropy methods for missing data; see, for instance, [11] and [12]); (ii) chemistry (e.g., the molecular conformation problem [5]); (iii) numerical analysis (e.g., optimization [17]); (iv) electrical engineering (e.g., data transmission, coding and image enhancement; see, for instance, [3]); and (v) geophysics (seismic reconstruction problems [13]).…”

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

“…. 1 DetA and 3x3 principal minors of A in Case 1 Table 4.1 DetA and 3x3 principal minors of A in Case 1 Table 4.2 DetA and 3x3 principal minors of A in Case 2 with e = 0 23 Table 4. 3 DetA and 3x3 principal minors of A in Case 3A Table 4.…”

“…A permutation similarity of A is a product PAP~ 1 where P is a permutation matrix. A diagonal similarity of A is a product DAD" 1 where D is a diagonal matrix.…”

“…in which all its diagonal entries are equal to 1. If the matrix A' has IT-completion, then so does the matrix A, because we can multiply the completed matrix A' on the left by the matrix I) 1 to obtain a completion of the matrix A. Since all the classes we consider have positive diagonal elements, we assume that the diagonal elements of every matrix are equal to 1.…”