Topic
Spectrum of a matrix
About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.
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TL;DR: In this article, a lower bound for the maximum eigenvalue of symmetric positive semidefinite matrices was obtained for the non-symmetric case, where the angle of the matrix is known.
21 citations
TL;DR: In this article, the eigenvalues w(m) of the grand canonical density matrix decay rapidly with their index m, like w (m)∼exp[−βB−1(ln m)1+1/α], where B and α are positive constants, O(1), which may be computed from the spectrum of the Hamiltonian.
Abstract: Using physical arguments and partition theoretic methods, we demonstrate under general conditions, that the eigenvalues w(m) of the grand canonical density matrix decay rapidly with their index m, like w(m)∼exp[−βB−1(ln m)1+1/α], where B and α are positive constants, O(1), which may be computed from the spectrum of the Hamiltonian. We compute values of B and α for several physical models, and confirm our theoretical predictions with numerical experiments. Our results have implications in a variety of questions, including the behaviour of fluctuations in ensembles, and the convergence of numerical density matrix renormalization group techniques.
21 citations
TL;DR: In this article, a novel solution strategy of Saint-Venant's problem based on Hamilton's formalism is proposed, which relies upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues.
Abstract: This paper proposes a novel solution strategy of Saint-Venant’s problem based on Hamilton’s formalism. Saint-Venant’s problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system’s Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant’s solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.
21 citations
TL;DR: In this article, the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specied eigenvalues was studied.
Abstract: We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specied eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specied region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in [Boutry et al. 2005] regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of BFGS and Lipschitz-based global optimization algorithms.
21 citations
Journal Article•
TL;DR: In this paper, the existence of sets that simultaneously tile R n by applying powers of an invertible matrix and translations by a lattice is studied, and sufficient and necessary conditions on the dilations and translations for the existence such sets are also given.
Abstract: Sets which simultaneously tile R n by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice rather than the usual dependence on the eigenvalues. For example,it is shown that for any values |a| > 1 > |b|,there is a (2 ×2) matrix A with eigenvalues a and b for which such a set exists,and a matrix A with eigenvalues a and b for which no such set exists. Finally,these results are related to the existence of wavelets for non-expansive dilations.
21 citations