Spectral conditioning and pseudospectral growth

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra. J. V. Burke’s research was supported in part by National Science Foundation Grant DMS-0505712. A. S. Lewis’s research was supported in part by National Science Foundation Grant DMS-0504032. M. L. Overton’s research was supported in part by National Science Foundation Grant DMS-0412049.     

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices. A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu.     

On Representations of Affine Hecke Algebras of Type B

Ariki’s and Grojnowski’s approach to the representation theory of affine Hecke algebras of type A is applied to type B with unequal parameters to obtain – under certain restrictions on the eigenvalues of the lattice operators – analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type A. Research supported by the Studienstiftung des deutschen Volkes.     

A Brauer’s theorem and related results

Given a square matrix A, a Brauer’s theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt’s and Hotelling’s deflations. An extension of the aforementioned Brauer’s result, Rado’s theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem.     

Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α ²-dynamo and circular string demonstrates the efficiency and applicability of the approach.     

The continuous spectrum for the M / M / c queue

Transient solutions for M/M/c queues are important for many purposes, in particular for staffing facilities such as call centers. In this article, we show how to use spectral analysis to find such solutions. The difficulty is that unless the number in line is bounded, one has to deal with matrices of infinite size, and hence with a countable infinite number of eigenvalues. This problem can be overcome by noting that the spectrum is dense with few exceptions. We also show how many discrete eigenvalues remain. Our theory may also work to obtain spectra for other infinite-dimensional matrices. Numerical properties of our approach are explored.     

The extension of the first main theorem for π-blocks

Let π be a set of primes and G a π-separable group. Isaacs defines the B _π characters, which can be viewed as the “π-modular” characters in G, such that the B _p′ characters form a set of canonical lifts for the p-modular characters. By using Isaacs’ work, Slattery has developed some Brauer’s ideals of p-blocks to the π-blocks of a finite π-separable group, generalizing Brauer’s three main theorems to the π-blocks. In this paper, depending on Isaacs’ and Slattery’s work, we will extend the first main theorem for π-blocks.     

On <Emphasis Type="Italic">Saaty</Emphasis>’s and <Emphasis Type="Italic">Koczkodaj</Emphasis>’s inconsistencies of pairwise comparison matrices

The aim of the paper is to obtain some theoretical and numerical properties of Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices (PRM). In the case of 3 ×  3 PRM, a differentiable one-to-one correspondence is given between Saaty’s inconsistency ratio and Koczkodaj’s inconsistency index based on the elements of PRM. In order to make a comparison of Saaty’s and Koczkodaj’s inconsistencies for 4  ×  4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n ×  n PRM is formulated, the elements a _ij (i < j) of which were randomly chosen from the ratio scale

Trace formulae forS 1 invariant green’s operators onS 2

Summary We study the spectrum of the Laplacian acting on 1-forms for a surface of revolution diffeomorphic toS ² and obtain, for theS ¹ invariant spectrum, a trace formula for the eigenvalues of its Green’s operator. The trace formula is used to “hear” negative curvature for some metrics and to prove the existence of metrics onS ² with rather unusual spectral characteristics. This article was processed by the author using the L^AT_EX style file from Springer-Verlag.     

K-area, Hofer metric and geometry of conjugacy classes in Lie groups

Given a closed symplectic manifold (M,ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M,ω) by means of the Hofer metric on Ham (M,ω). We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold (M,ω) to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with boundary in the spirit of L. Polterovich’s work on Hamiltonian fibrations over S ². Oblatum 23-II-2001 & 9-V-2001?Published online: 20 July 2001     

PARTIAL FERRERS MATRICES

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Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties

Batyrev et al. constructed a family of Calabi–Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anticanonical hypersurfaces in G/B. Recently, Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev’s in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of degeneration. The birational maps involved are closely related to Berenstein and Zelevinsky’s geometric lifting of tropical maps to maps between totally positive varieties.     

Principal eigenvalues,topological pressure,and stochastic stability of equilibrium states

Suppose thatL is a second order elliptic differential operator on a manifoldM, B is a vector field, andV is a continuous function. The paper studies by probabilistic and dynamical systems means the behavior asɛ → 0 of the principal eigenvalueλ ^ε (V) for the operatorL ^ε = ɛL + (B, ∇) +V considered on a compact manifold or in a bounded domain with zero boundary conditions. Under certain hyperbolicity conditions on invariant sets of the dynamical system generated by the vector fieldB the limit asɛ → 0 of this principal eigenvalue turns out to be the topological pressure for some function. This gives a natural transition asɛ → 0 from Donsker-Varadhan’s variational formula for principal eigenvalues to the variational principle for the topological pressure and unifies previously separate results on random perturbations of dynamical systems. This work was supported by US-Israel Binational Science Foundation.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

A generalization of weyl’s inequalities with implications

This paper suggests a generalization of the additive Weyl inequalities to the case of two square matrices of different orders. As a consequence of the generalized Weyl inequalities, a theorem describing the location of eigenvalues of a Hermitian matrix in terms of the eigenvalues of an arbitrary Hermitian matrix of smaller order is derived. It is demonstrated that the latter theorem provides a generalization of Kahan’s theorem on clustered eigenvalues. It is also shown that the theorem on extended interlacing intervals is another consequence of the generalized additive Weyl inequalities suggested. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 49–59. Translated by L. Yu. Kolotilina.     

B-fredholm and spectral properties for multipliers in Banach algebras

The main purpose of this paper is to study spectral and B-Fredholm properties of a multiplierT acting on a semi-simple regular tauberian commutative Banach algebraA. We show thatT is a B-Fredholm operator if and only ifT is a semi B-Fredholm operator, and in this case we have the indexind(T)=0. Moreover we give some spectral properties for multipliers. Spectral mapping theorems for the Weyl’s and B-Weyl spectrum of a multiplier are also considered. Furthermore we show that Weyl’s theorem and generalized Weyl’s theorem hold for a multiplierT. Finally we give sufficient conditions for a multiplier to be a product of an invertible and an idempotent operators.     

Posterior two-sided estimates for eigenvalues of the singular Sturm-Liouville problem

Based on three-point difference and variational-difference schemes for auxiliary nonsingular spectral problems providing for a two-sided approximation of eigenvalues of the singular Sturm-Liouville problem, posterior upper and lower estimates for eigenvalues of the input singular problem are obtained. Bibliography: 3 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 39–49.     

Computing Eigenelements of Real Symmetric Matrices via Optimization

In certain circumstances, it is advantageous to use an optimization approach in order to solve the generalized eigenproblem, Ax = Bx, where A and B are real symmetric matrices and B is positive definite. In particular, this is the case when the matrices A and B are very large the computational cost, prohibitive, of solving, with high accuracy, systems of equations involving these matrices. Usually, the optimization approach involves optimizing the Rayleigh quotient.We first propose alternative objective functions to solve the (generalized) eigenproblem via (unconstrained) optimization, and we describe the variational properties of these functions.We then introduce some optimization algorithms (based on one of these formulations) designed to compute the largest eigenpair. According to preliminary numerical experiments, this work could lead the way to practical methods for computing the largest eigenpair of a (very) large symmetric matrix (pair).     

The computation of isotropic vectors

We describe algorithms to compute isotropic vectors for matrices with real or complex entries. These are unit vectors b satisfying b ^* Ab = 0. For real matrices the algorithm uses only the eigenvectors of the symmetric part corresponding to the extreme eigenvalues. For complex matrices, we first use the eigenvalues and eigenvectors of the Hermitian matrix K = (A − A ^*)/2i. This works in many cases. In case of failure we use the Hermitian part H or a combination of eigenvectors of H and K. We give some numerical experiments comparing our algorithms with those proposed by R. Carden and C. Chorianopoulos, P. Psarrakos and F. Uhlig.     

Mathematics of Clifford - a Maple package for Clifford and Graßmann algebras

CLIFFORD performs various computations in Gra?mann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C ℓ (B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUM - based on Chevalley’s recursive formula, and cmulRS - based on a non-recursive Rota-Stein sausage. Gra?mann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.