Lower bounds for the largest eigenvalue of the gcd matrix on {1, 2,..., n}

被引：1

作者：

Merikoski, Jorma K. ^{[1
]}

机构：

[1] Univ Tampere, Sch Informat Sci, Kanslerinrinne 1, FI-33014 Tampere, Finland

来源：

CZECHOSLOVAK MATHEMATICAL JOURNAL | 2016年 / 66卷 / 03期

关键词：

eigenvalue bounds; greatest common divisor matrix; COMMON DIVISOR MATRICES;

D O I：

10.1007/s10587-016-0307-5

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Consider the nxn matrix with (i, j)'th entry gcd (i, j). Its largest eigenvalue lambda (n) and sum of entries s (n) satisfy lambda (n) > s (n) /n. Because s (n) cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that lambda (n) > 6 pi(-2) nlogn for all n. If n is large enough, this follows from F.Balatoni (1969).

引用

页码：1027 / 1038

页数：12

共 50 条

[21] Some bounds for the largest eigenvalue of a signed graph
Stanic, Zoran
[J]. BULLETIN MATHEMATIQUE DE LA SOCIETE DES SCIENCES MATHEMATIQUES DE ROUMANIE, 2019, 62 (02): : 183 - 189
[22] Lower bounds on eigenvalue summation for the solution of the Lyapunov matrix differential equation
Zhang, Juan
Liu, Jianzhou
Li, Quanbing
[J]. IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION, 2017, 34 (03) : 987 - 998
[23] Lower Eigenvalue Bounds on Summation for the Solution of the Lyapunov Matrix Differential Equation
Zhang, Juan
Liu, Jianzhou
Huang, Hao
[J]. ASIAN JOURNAL OF CONTROL, 2017, 19 (01) : 382 - 390
[24] ON LOWER BOUNDS FOR THE SMALLEST EIGENVALUE OF A HERMITIAN POSITIVE-DEFINITE MATRIX
MA, EM
ZAROWSKI, CJ
[J]. IEEE TRANSACTIONS ON INFORMATION THEORY, 1995, 41 (02) : 539 - 540
[25] Lower bounds of the minimal eigenvalue of a Hermitian positive-definite matrix
Sun, WW
[J]. IEEE TRANSACTIONS ON INFORMATION THEORY, 2000, 46 (07) : 2760 - 2762
[26] On the largest eigenvalue of the distance matrix of a tree
Zhou, Bo
[J]. MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY, 2007, 58 (03) : 657 - 662
[27] ON MINIMIZING THE LARGEST EIGENVALUE OF A SYMMETRICAL MATRIX
FAN, MKH
NEKOOIE, B
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 1995, 214 : 225 - 246
[28] On the largest eigenvalue of extended eccentric matrix
Ghorbani, Modjtaba
Alidehi-Ravandi, Razie
Amraei, Najaf
Raza, Zahid
[J]. CHAOS SOLITONS & FRACTALS, 2024, 185
[29] Eigenvalue upper and lower bounds of the solution for the continuous algebraic matrix Riccati equation
Lee, CH
[J]. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS, 1996, 43 (08): : 683 - 686
[30] Bounds for the largest p-Laplacian eigenvalue for graphs
Amghibech, S.
[J]. DISCRETE MATHEMATICS, 2006, 306 (21) : 2762 - 2771

← 1 2 3 4 5 →