Lower Bounds for the Rank of a Matrix with Zeros and Ones outside the Leading Diagonal

We found a lower bound on the rank of a square matrix where every entry in the leading diagonal is neither zero nor one and every entry outside the leading diagonal is either zero or one. The rank of this matrix is at least half its order. Under an additional condition, the lower bound is higher by...

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Published inProgramming and computer software Vol. 50; no. 2; pp. 202 - 207
Main Authors Seliverstov, A. V., Zverkov, O. A.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.04.2024
Springer Nature B.V
Subjects
Algebra
Artificial Intelligence
Computer algebra
Computer Science
Estimates
Hypotheses
Independent variables
Linear algebra
Linear equations
Lower bounds
Matrices (mathematics)
Matrix algebra
Operating Systems
Polytopes
Prime numbers
Software Engineering
Software Engineering/Programming and Operating Systems
Variables
matrix rank
system of linear equations
and SymPy
computer algebra system
Online AccessGet full text
ISSN0361-7688
1608-3261
DOI10.1134/S0361768824020142

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Summary:We found a lower bound on the rank of a square matrix where every entry in the leading diagonal is neither zero nor one and every entry outside the leading diagonal is either zero or one. The rank of this matrix is at least half its order. Under an additional condition, the lower bound is higher by one. This condition means that some auxiliary system of linear equations has no binary solution. Some examples are provided that show that the lower bound can be achieved. This lower bound on the matrix rank allows the problem of finding a binary solution to a system of linear equations with a sufficiently large number of linearly independent equations to be reduced to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each differing from the binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a large system of linear algebraic equations. Estimates of the time required for calculating the matrix rank in the SymPy computer algebra system are also provided. It is shown that the rank of a matrix over the field of residues modulo prime number is calculated faster than it generally takes to calculate the rank of a matrix of the same order over the field of rational numbers.
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ISSN:0361-7688
1608-3261
DOI:10.1134/S0361768824020142