A new inclusion interval for the real eigenvalues of real matrices

Yinghua Wang; Xinnian Song; Lei Gao

Displaying similar documents to “A new inclusion interval for the real eigenvalues of real matrices”

Complexity of computing interval matrix powers for special classes of matrices

David Hartman, Milan Hladík (2020)

Applications of Mathematics

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Computing powers of interval matrices is a computationally hard problem. Indeed, it is NP-hard even when the exponent is 3 and the matrices only have interval components in one row and one column. Motivated by this result, we consider special types of interval matrices where the interval components occupy specific positions. We show that computing the third power of matrices with only one column occupied by interval components can be solved in cubic time; so the asymptotic time complexity...

Nested matrices and inverse $M$ -matrices

Jeffrey L. Stuart (2015)

Czechoslovak Mathematical Journal

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Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the $L U$ - and $Q R$ -factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse $M$ -matrices with symmetric, irreducible, tridiagonal inverses.

Eigenvalue bounds for some classes of matrices associated with graphs

Ranjit Mehatari, M. Rajesh Kannan (2021)

Czechoslovak Mathematical Journal

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For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first, we derive bounds for the second largest and the smallest eigenvalues of adjacency matrices of $k$ -regular graphs. Then we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest and the largest eigenvalues of the Laplacian matrices of graphs. The sharpness of these bounds...

An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices

Assaf Goldberger, Neumann, Michael (2004)

Czechoslovak Mathematical Journal

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Suppose that $A$ is an $n \times n$ nonnegative matrix whose eigenvalues are $λ = ρ (A), λ_{2}, ..., λ_{n}$ . Fiedler and others have shown that $det (λ I - A) \leq λ^{n} - ρ^{n}$ , for all $λ > ρ$ , with equality for any such $λ$ if and only if $A$ is the simple cycle matrix. Let $a_{i}$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i \times i$ , $i = 1, ..., n - 1$ . We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $det (λ I - A) + \sum_{i = 1}^{n - 1} ρ^{n - 2 i} | a_{i} | {(λ - ρ)}^{i} \leq λ^{n} - ρ^{n}$ , for all $λ \geq ρ$ . We use this inequality to derive the inequality that: $\prod_{2}^{n} (ρ - λ_{i}) \leq ρ^{n - 2} \sum_{i = 2}^{n} (ρ - λ_{i})$ . In the spirit of a celebrated conjecture...

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

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Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) $\sum_{i = 1}^{n} X ₂_{i} - d \sum_{i = 1}^{n} Y ²_{i} = - I$ with d ∈ N for nonsingular $X_{i}, Y_{i} \in M ₂ (Z)$ , i=1,...,n.

Absolute continuity for Jacobi matrices with power-like weights

Wojciech Motyka (2007)

Colloquium Mathematicae

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This work deals with a class of Jacobi matrices with power-like weights. The main theme is spectral analysis of matrices with zero diagonal and weights $λ ₙ : = n^{α} (1 + Δ ₙ)$ where α ∈ (0,1]. Asymptotic formulas for generalized eigenvectors are given and absolute continuity of the matrices considered is proved. The last section is devoted to spectral analysis of Jacobi matrices with qₙ = n + 1 + (-1)ⁿ and $λ ₙ = \sqrt (q_{n - 1} q ₙ)$ .

Inverse eigenvalue problem for constructing a kind of acyclic matrices with two eigenpairs

Maryam Babaei Zarch, Seyed Abolfazl Shahzadeh Fazeli, Seyed Mehdi Karbassi (2020)

Applications of Mathematics

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We investigate an inverse eigenvalue problem for constructing a special kind of acyclic matrices. The problem involves the reconstruction of the matrices whose graph is an $m$ -centipede. This is done by using the $(2 m - 1)$ st and $(2 m)$ th eigenpairs of their leading principal submatrices. To solve this problem, the recurrence relations between leading principal submatrices are used.

A conjugacy theorem for subgroups of $G L_{n}$ containing the group of diagonal matrices

N. A. Vavilov (1987)

Colloquium Mathematicae

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A matrix formalism for conjugacies of higher-dimensional shifts of finite type

Michael Schraudner (2008)

Colloquium Mathematicae

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We develop a natural matrix formalism for state splittings and amalgamations of higher-dimensional subshifts of finite type which extends the common notion of strong shift equivalence of ℤ⁺-matrices. Using the decomposition theorem every topological conjugacy between two $ℤ^{d}$ -shifts of finite type can thus be factorized into a finite chain of matrix transformations acting on the transition matrices of the two subshifts. Our results may be used algorithmically in computer explorations on...

Theorems of the alternative for cones and Lyapunov regularity of matrices

Bryan Cain, Daniel Hershkowitz, Hans Schneider (1997)

Czechoslovak Mathematical Journal

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Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^{*}$ and $D^{*}$ may overlap. When $T V \to W$ is linear and $K \subset V$ and $D \subset W$ are cones, these results will be applied to $C = T (K)$ and $D$ , giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$ . The case when $V = W$ is the space $H$ of $n \times n$ Hermitian matrices, $D$ is the $n \times n$ positive semidefinite matrices, and $T (X) = A X + X^{*} A$ yields new and known results about the existence...

A method to rigorously enclose eigenpairs of complex interval matrices

Castelli, Roberto, Lessard, Jean-Philippe

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In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair $x = (λ,)$ is found by solving a nonlinear equation of the form $f (x) = 0$ via a contraction argument. The set-up of the method relies on the notion of $r a d i i p o l y n o m i a l s$ , which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.

Eigenvalues and eigenvectors of some tridiagonal matrices with non-constant diagonal entries

S. Kouachi (2008)

Applicationes Mathematicae

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We give explicit expressions for the eigenvalues and eigenvectors of some tridiagonal matrices with non-constant diagonal entries. Our techniques are based on the theory of recurrent sequences.