Bounds for characteristic values of positive definite matrices (original) (raw)

Bounds for eigenvalues using the trace and determinant

Linear Algebra and its Applications, 1997

Let A be a square matrix with real and positive eigenvalues A 1 >/ --->1 A n > 0, and let 1 ~< k ~< I ~< n. Bounds for A k "-A l and A k + "" + A t, involving k, 1, n, tr A, and det A only, are presented. @ 1997 Elsevier Seienee Inc.

Improving eigenvalue bounds using extra bounds

Linear Algebra and its Applications, 1985

can be improved by combining them with other bounds. In particular, the diagonal of A, in conjunction with majorization, is used to improve the bounds. These bounds all require 0( n2) multiplications.

Computing a Lower Bound of the smallest Eigenvalue of a Symmetric Positive Definite

In this paper several algorithms to compute a lower bound of the smallest eigenvalue of a symmetric positive definite Toeplitz matrix are described and compared in terms of accuracy and computational effi- ciency. Exploiting the Toeplitz structure of the considered matrix, new the- oretical insights are derived and an efficient implementation of some of the aforementioned algorithms is provided. AMS Subject Classification: 65F15

Bounds for sums of eigenvalues and applications

Computers & Mathematics with Applications, 2000

Let A be a matrix of order n × n with real spectrum A 1 ~ A2 ~_ " • • _~> An. Let 1 < k < n -2. If An or A1 is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remaining eigenwlues of A. Then, we obtain a majorization vector for (A1, A2,..., An-l) when An is known and a majorization vector for (A2, A3,..., An) when A1 is known. We apply these results to the eigenvalues of the Laplacian matrix of a graph and, in particular, a sufficient condition for a graph to be connected is given. Also, we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum. (~)

New bounds for the Eigenvalues of Matrix Polynomials

European Journal of Pure and Applied Mathematics

We employ several numerical radius inequalities to the square of the Frobenius companion matrices of monic matrix polynomials to provide new bounds for the eigenvalues of these polynomials.

On the positive definiteness and eigenvalues of

2012

In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set S n is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix (S n) f. From this condition we obtain some sufficient conditions for positive definiteness as corollaries. We also use graph theory and show that by making some graph theoretic assumptions on the set S n we are able to reduce the assumptions on the function f while still preserving the positive definiteness of the matrix (S n) f. Dual theorems of these results for join matrices are also presented. As examples we consider the so-called power GCD and power LCM matrices as well as MIN and MAX matrices. Finally we give bounds for the eigenvalues of meet and join matrices in cases when the function f possesses certain monotonic behaviour.