On GCD and LCM matrices

DOI10.1016/0024-3795(92)90042-9zbMath0761.15013OpenAlexW1995734876MaRDI QIDQ1195336

Steve Ligh, Keith Bourque

Publication date: 26 October 1992

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0024-3795(92)90042-9

zbMATH Keywords

determinantgreatest common divisor matrixleast common multiple matrixintegral matrix

Mathematics Subject Classification ID

Determinants, permanents, traces, other special matrix functions (15A15) Matrices of integers (15B36) Matrices, determinants in number theory (11C20)

Related Items (49)

Divisibility among determinants of power matrices associated with integer-valued arithmetic functions ⋮ Determinants of Smith matrices on three coprime divisor chains and divisibility ⋮ Determinants of matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains ⋮ On the factorization of LCM matrices on gcd-closed sets ⋮ Divisibility among power matrices associated with classes of arithmetic functions ⋮ Divisibility among power matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains ⋮ Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions ⋮ On Smith's determinant ⋮ On the divisibility among power GCD and power LCM matrices on gcd-closed sets ⋮ Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains ⋮ Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains ⋮ Matrices associated with arithmetical functions ⋮ Upper bounds for the condition numbers of the GCD and the reciprocal GCD matrices in spectral norm ⋮ On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices ⋮ THERE DOES NOT EXIST AN ODD SINGULAR NUMBER OF THE FORM p^lqr ⋮ Divisibility of matrices associated with multiplicative functions ⋮ On the positive definiteness and eigenvalues of meet and join matrices ⋮ Determinants and divisibility of power GCD and power LCM matrices on finitely many coprime divisor chains ⋮ Notes on Hong's conjectures of real number power LCM matrices ⋮ Studying the singularity of LCM-type matrices via semilattice structures and their Möbius functions ⋮ On the divisibility of meet and join matrices ⋮ Divisibility properties of power GCD matrices and power LCM matrices ⋮ On the additive complexity of GCD and LCM matrices ⋮ Explicit expressions of Smith's determinant on a poset ⋮ A proof of a conjecture on monotonic behavior of the smallest and the largest eigenvalues of a number theoretic matrix ⋮ Nonsingularity of matrices associated with classes of arithmetical functions ⋮ Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains ⋮ Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets ⋮ Divisibility of LCM matrices by totally nonnegative GCD matrices ⋮ Non-divisibility of LCM matrices by GCD matrices on \(\gcd\)-closed sets ⋮ On inverses of GCD matrices associated with multiplicative functions and a proof of the Hong-Loewy conjecture ⋮ On the divisibility of power LCM matrices by power GCD matrices ⋮ On Hong’s conjecture for power LCM matrices ⋮ Matrices associated with multiplicative functions ⋮ Studying the inertias of LCM matrices and revisiting the Bourque-Ligh conjecture ⋮ Bounds for determinants of matrices associated with classes of arithmetical functions ⋮ Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains ⋮ GCD-closed sets and divisibility of Smith matrices ⋮ A lattice-theoretic approach to the Bourque–Ligh conjecture ⋮ Divisibility of power LCM matrices by power GCD matrices on gcd-closed sets ⋮ Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets ⋮ On meet matrices on posets ⋮ Nonsingularity of least common multiple matrices on gcd-closed sets ⋮ On Nonsingular Power LCM Matrices ⋮ Some analogues of smith's determinant ⋮ On the Bourque-Ligh conjecture of least common multiple matrices ⋮ Reciprocal power GCDQ matrices and power LCMQ matrices defined on factor closed sets over Euclidean domains ⋮ A topological approach to divisibility of arithmetical functions and GCD matrices ⋮ ON THE DIVISIBILITY AMONG POWER LCM MATRICES ON GCD-CLOSED SETS

Cites Work

• The determinants of GCD matrices
• Greatest common divisor matrices
• Higher-dimensional gcd matrices
• Arithmetical properties of generalized Ramanujan sums
• A Generalization of Smith's Determinant
• Another generalisation of smith's determinant
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