Nonsingular ACI-matrices over integral domains

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A partial matrix over a field F is a matrix whose entries are either an element of F or an indeterminate and with each indeterminate only appearing once. A completion is an assignment of values in F to all indeterminates. Given a partial matrix, through elementary row operations and column permutation it can be decomposed into a block matrix of the formwhere W is wide (has more columns than rows), S is square, T is tall (has more rows than columns), and these three blocks have at least one completion with full rank. And importantly, each one of the blocks W, S and T is unique up to elementary row operations and column permutation whenever S is required to be as large as possible. When this is the case W * * 0 S * 0 0 T will be called a WST-decomposition. With this decomposition it is trivial to compute maximum rank of a completion of the original partial matrix: rows(W) + rows(S) + cols(T). In fact we introduce the WST-decomposition for a broader class of matrices: the ACI-matrices.

Section: Preliminariesmentioning

confidence: 99%

The WST-decomposition for partial matrices

Borobia

Canogar

2019

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“…, V n } by the m × n ACI-matrix where the column j corresponds to the affine subspace V j . A completion A of the ACI-matrix A given in (1) is an assignment of values in F to each one of the indeterminates x…”

Section: A Geometric Interpretationmentioning

confidence: 99%

“…In [1] we proved that item (i) is true for any field F without the restriction |F| ≥ max{m, n + 1}. In [2, Lemma 2.1 and Lemma 3.1] we proved the existence of minimal full rank ACI-matrices and of maximal full rank ACI-matrices over all finite fields (see Example 1.3), and we showed that if A is minimal full rank and B is maximal full rank then…”

Section: Theorem 24 (Detailed Version)mentioning

confidence: 99%

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ACI-matrices of constant rank over arbitrary fields

Borobia

Canogar

2017

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The columns of a m × n ACI-matrix over a field F are independent affine subspaces of F m . An ACI-matrix has constant rank ρ if all its completions have rank ρ. Huang and Zhan (2011) characterized the m × n ACI-matrices of constant rank when |F| ≥ min{m, n + 1}. We complete their result characterizing the m × n ACI-matrices of constant rank over arbitrary fields. Quinlan and McTigue (2014) proved that every partial matrix of constant rank ρ has a ρ × ρ submatrix of constant rank ρ if and only |F| ≥ ρ. We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility. *

“…The more general situation of partial matrices whose completions have ranks satisfying a specified upper bound was investigated by Cohen et al in [4], and this investigation was extended to ACI-matrices in [2]. Some of the results of [2] have recently been extended to the case of partial and ACI-matrices over arbitrary integral domains by Borobia and Canogar [1].…”

Section: Introductionmentioning

confidence: 99%

Partial matrices whose completions all have the same rank

McTigue

Quinlan

2013

A partial matrix over a field F is a matrix whose entries are either elements of F or independent indeterminates. A completion of such a partial matrix is obtained by specifying values from F for the indeterminates. We determine the maximum possible number of indeterminates in an m × n partial matrix (m ≤ n) whose completions all have a particular rank r, and we fully describe those examples in which this maximum is attained, without any restriction on the field F.