A lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$-matrix and its inverse

Wenlong Zeng; Jianzhou Liu

Displaying similar documents to “A lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse”

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an $2 n \times 2 n$ Brualdi-Li tournament matrix $B_{2 n}$ . We show that $B_{2 n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2 n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.

On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Yusaku Yamamoto (2017)

Applications of Mathematics

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Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix $A \in ℝ^{m \times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $Tr (A^{- 1})$ and $Tr (A^{- 2})$ have attracted attention recently, because they can be computed in $O (m)$ operations when $A$ is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that...

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

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

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Consider the $n \times n$ matrix with $(i, j)$ ’th entry $gcd (i, j)$ . Its largest eigenvalue $λ_{n}$ and sum of entries $s_{n}$ satisfy $λ_{n} > s_{n} / n$ . 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 $λ_{n} > 6 π^{- 2} n log n$ for all $n$ . If $n$ is large enough, this follows from F. Balatoni (1969).

Partial sum of eigenvalues of random graphs

Israel Rocha (2020)

Applications of Mathematics

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Let $G$ be a graph on $n$ vertices and let $λ_{1} \geq λ_{2} \geq ... \geq λ_{n}$ be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues $s_{k} = \sum_{i = 1}^{k} λ_{i}$ , for $1 \leq k \leq n$ , and show that a typical graph has $s_{k} \leq (e (G) + k^{2}) / {(0.99 n)}^{1 / 2}$ , where $e (G)$ is the number of edges of $G$ . We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix.

Nonlinear mappings preserving at least one eigenvalue

Constantin Costara, Dušan Repovš (2010)

Studia Mathematica

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We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F (x) = u x u^{- 1}$ or $F (x) = u x^{t} u^{- 1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...

The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula

Chein-Shan Liu, Botong Li (2024)

Applications of Mathematics

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The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an $n$ -dimensional matrix eigenvalue problem is derived with a special matrix $𝐀 : = [a_{i j}]$ , that is, $a_{i j} = 0$ if $i + j$ is odd.Based on the product formula, an integration method with a fictitious...

A spectral bound for graph irregularity

Felix Goldberg (2015)

Czechoslovak Mathematical Journal

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The imbalance of an edge $e = {u, v}$ in a graph is defined as $i (e) = | d (u) - d (v) |$ , where $d (\cdot)$ is the vertex degree. The irregularity $I (G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$ . This concept was introduced by Albertson who proved that $I (G) \leq 4 n^{3} / 27$ (where $n = | V (G) |$ ) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...

Analytic aspects of the circulant Hadamard conjecture

Teodor Banica, Ion Nechita, Jean-Marc Schlenker (2014)

Annales mathématiques Blaise Pascal

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We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $| q_{0} | = ... = | q_{N - 1} | = 1$ the quantity $Φ = \sum_{i + k = j + l} \frac{q_{i} q_{k}}{q_{j} q_{l}}$ satisfies $Φ \geq N^{2}$ , with equality if and only if $q = (q_{i})$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $Φ$ , (2) the study of the critical points of $Φ$ , and (3) the computation of the moments of $Φ$ . We explore here...

Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

Dario Fasino, Francesco Tudisco (2016)

Czechoslovak Mathematical Journal

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We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$ . The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$ . Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show that the choice where $X$ is the all-ones matrix allows to estimate the signature of the leading eigenvector of $A$ , generalizing previous results on Perron-Frobenius properties of matrices...

A Fiedler-like theory for the perturbed Laplacian

Israel Rocha, Vilmar Trevisan (2016)

Czechoslovak Mathematical Journal

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The perturbed Laplacian matrix of a graph $G$ is defined as $L^{D} = D - A$ , where $D$ is any diagonal matrix and $A$ is a weighted adjacency matrix of $G$ . We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore,...

Displaying similar documents to “A lower bound sequence for the minimum eigenvalue of Hadamard product of an M -matrix and its inverse”

Displaying similar documents to “A lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse”