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In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A ( k ) converge to zero. However, this fact alone does not necessarily guarantee that A ( k ) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q ( k ) . We prove that under certain assumptions A ( k ) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q ( k ) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q ( k ) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example.

The change in geometric multiplicity, associated with perturbing a particular pair of off-diagonal entries, in a combinatorially symmetric matrix is investigated. First, we focus upon a Hermitian matrix, with a general graph, when the perturbation is Hermitian. This generalizes prior work in case the graph is a tree; the possibilities are much richer. Then, we turn to general matrices over a field. In both cases, by classifying the incident vertices, all possibilities for the change in geometric multiplicity are identified. Then, examples are given to show that each possibility may actually occur. In some instance, the change in geometric multiplicity depends upon the numerical value of the perturbation, and it matters that both off-diagonal entries change. However, in most instances, the change is qualitatively determined.

In this paper, we study T-eigenvalues of third-order tensors. Definitions of the T-eigenvalues and Hermitian tensors are proposed. We present a commutative tensor family. We prove some T-eigenvalue inequalities for Hermitian tensors, including extensions of Weyl's theorem and Cauchy's interlacing theorem from the matrix case to the tensor case. Finally, we introduce the stability of the T-eigenvalues and study the Lyapunov equation for tensors.

Let $A$ be an $n \times n$ positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of $A$ as the integral of some explicit log-concave function on ${\Bbb R}^{2n}$. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of $A$., 5 pages

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.

The singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E ( A ) is the sum of its singular values. The energy of a graph G, E ( G ) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E ( A ) ≥ 2 E ( D ) . Also, we show that if G is a graph and H is a spanning subgraph of G such that E ( H ) is an edge cut of G, then E ( H ) ≤ E ( G ) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E ( G ) ≥ 4 m + n ( n − 2 ) | det ⁡ ( A ) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E ( G ) given in Oboudi (2019) [14] .

We study certain pairs of subspaces V and W of C^n we call supports that consist of eigenspaces of the eigenvalues ±‖M‖ of a minimal hermitian matrix M(‖M‖ ≤‖M+D‖ for all real diagonals D). For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds. Fil: Mendoza, Alberto. Universidad Simón Bolívar; Venezuela Fil: Recht, Lázaro. Universidad Simón Bolívar; Venezuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

This paper is devoted to the asymptotic behavior of all eigenvalues of the increasing finite principal sections of an infinite symmetric (in general non-Hermitian) Toeplitz matrix. The symbol of the infinite matrix is supposed to be moderately smooth and to trace out a simple loop in the complex plane. The main result describes the asymptotic structure of all eigenvalues. The asymptotic expansions constructed are uniform with respect to the location of the eigenvalues in the bulk of the spectrum.

The FEAST algorithm, a contour-integral based eigensolver, was developed for computing the eigenvalues inside a given interval, along with their eigenvectors, of a Hermitian generalized eigenproblem. The FEAST algorithm is a fast and stable technique, and is easily parallelizable. However, it was shown that this algorithm may fail to find the desired eigenpairs when applied to non-Hermitian problems. Efforts have been made to adapt FEAST to non-Hermitian problems. In this work, we aim at formulating a new non-Hermitian scheme for the FEAST algorithm. Our new scheme is based on the partial generalized Schur decomposition. Unlike the existing non-Hermitian FEAST methods, our method seeks approximation to the generalized Schur vectors instead of the potentially ill-conditioned eigenvectors. Our new method can compute not only the eigenvectors but also the right and left generalized Schur vectors corresponding to target eigenvalues. We present standard benchmark numerical experiments in which our method achieves better stability and efficiency comparing with the existing non-Hermitian FEAST algorithms.

It is known that an n-by-n Hermitian matrix, n ≥ 2 , whose graph is a tree necessarily has at least two eigenvalues (the largest and smallest, in particular) with multiplicity 1. Recently, much of the multiplicity theory, for eigenvalues of Hermitian matrices whose graph is a tree, has been generalized to geometric multiplicities of eigenvalues of matrices over a general field (whose graph is a tree). However, the two 1's fact does not generalize. Here, we give circumstances under which there are two 1's and give several examples (without two 1's) that limit our positive results.

A generalization of the method developed by Adam, Psarrakos and Tsatsomeros to find inequalities for the eigenvalues of a complex matrix A using knowledge of the largest eigenvalues of its Hermitian part H(A) is presented. The numerical range or field of values of A can be constructed as the intersection of half-planes determined by the largest eigenvalue of H($e^{i\theta}$A). Adam, Psarrakos and Tsatsomeros showed that using the two largest eigenvalues of H(A), the eigenvalues of A satisfy a cubic inequality and the envelope of such cubic curves defines a region in the complex plane smaller than the numerical range but still containing the spectrum of A. Here it is shown how using the three largest eigenvalues of H(A) or more, one obtains new inequalities for the eigenvalues of A and new envelope-type regions containing the spectrum of A., Comment: 19 pages, 11 figures. Version published in Lin. Alg. Appl. 557, 1-21, 2018

For the nonsingular generalized saddle-point matrix of a Hermitian positive definite or semidefinite leading block, we rigorously analyze clustering property for the eigenvalues of the corresponding preconditioned matrix with respect to the Hermitian and skew-Hermitian splitting preconditioner. The result shows that these eigenvalues are clustered around 0 + , 2 − , and a few points located on the unit circle centered at 1, as the iteration parameter is close to 0.

We characterize the possible lists of multiplicities occurring among the eigenvalues of real symmetric (or Hermitian) matrices whose graph is one of Kn, Kn less an edge, or both possibilities for Kn less two edges. The lists are quite different from those for trees. Some construction techniques are developed here and additional results with more missing edges are given, including the case of several independent edges., This research was financed by FEDER Funds through “Programa Operacional Factores de Competitividade - COMPETE” and by Portuguese Funds through FCT, within the Project PEst-OE/MAT/UI0013/2014., info:eu-repo/semantics/publishedVersion

For real symmetric, or complex Hermitian, matrices whose graph is a tree, there is a well-developed (via several papers) theory about the possible multiplicities of the eigenvalues. It includes a theory of vertices whose removal increases a multiplicity (the “Parter–Wiener, etc. theory” and the “downer branch mechanism”), how to determine maximum multiplicity, lower bounds for the minimum number of distinct eigenvalues, etc. Remarkably, a great deal of this theory may be generalized to geometric multiplicities of general matrices over a field (with very different proofs). We show here what parts of this theory generalize, and, in the process, review the theory.

The paper presents the information processing that can be performed by a general hermitian matrix when two of its distinct eigenvalues are coupled, such as λ λ ′ , instead of considering only one eigenvalue as traditional spectral theory does. Setting a = λ + λ ′ 2 ≠ 0 and e = λ ′ − λ 2 > 0 , the information is delivered in geometric form, both metric and trigonometric, associated with various right-angled triangles exhibiting optimality properties quantified as ratios or product of | a | and e. The potential optimisation has a triple nature which offers two possibilities: in the case λ λ ′ > 0 they are characterised by e | a | and | a | e and in the case λ λ ′ 0 by | a | e and | a | e . This nature is revealed by a key generalisation to indefinite matrices over R or C of Gustafson's operator trigonometry.

We obtain some generalisations of the inequalities for positive unital linear maps on the algebra of matrices. As a consequence, we obtain several positive semidefinite matrices and new eigenvalue inequalities for Hermitian matrices.

The computation of eigenvalues and eigenvectors of matrix polynomials is an important, but difficult, problem. The standard approach to solve this problem is to use linearizations, which are matrix polynomials of degree 1 that share the eigenvalues of P ( λ ) . Hermitian matrix polynomials and their real eigenvalues are of particular interest in applications. Attached to these eigenvalues is a set of signs called the sign characteristic. From both a theoretical and a practical point of view, it is important to be able to recover the sign characteristic of a Hermitian linearization of P ( λ ) from the sign characteristic of P ( λ ) . In this paper, for a Hermitian matrix polynomial P ( λ ) with nonsingular leading coefficient, we describe, in terms of the sign characteristic of P ( λ ) , the sign characteristic of the Hermitian linearizations in the vector space DL ( P ) (Mackey, Mackey, Mehl and Mehrmann, 2006). In particular, we identify the Hermitian linearizations in DL ( P ) that preserve the sign characteristic of P ( λ ) . We also provide a description of the sign characteristic of the Hermitian linearizations of P ( λ ) in the family of generalized Fiedler pencils with repetition (Bueno, Dopico, Furtado and Rychnovsky, 2015).

Let G be a mixed graph with n vertices, H ( G ) the Hermitian adjacency matrix of G , and λ 1 ( G ) , λ 2 ( G ) , … , λ n ( G ) the eigenvalues of H ( G ) . The Hermitian energy of G is defined as E H ( G ) = ∑ i = 1 n | λ i ( G ) | . In this paper we characterize the limiting spectral distribution of the Hermitian adjacency matrices of random mixed graphs, and as an application, we give an estimation of the Hermitian energy for almost all mixed graphs.

For an Hermitian matrix A whose graph is a tree T , we study the number of eigenvalues of A whose multiplicity decreases when a particular vertex is deleted from T . Explicit results are given when that number of eigenvalues is less than 4 and an inductive result thereafter. The work is based, in part, on classical results about multiplicities, but also on some new facts, including a useful identity. This allows us to give strong bounds based on simple facts about the location of the vertex in the tree. Some facts about matrices whose graphs are not trees are included, and the classical diameter bound about the number of distinct eigenvalues for a tree follows.

Let n ∈ N and [ A j k ] j , k = 1 , 2 be a Hermitian 2 n × 2 n matrix partitioned into four quadratic matrices A j k of order n. Marcus and Watkins proved in 1971 that [ tr A j k 2 ] is positive semidefinite whenever [ A j k ] is positive semidefinite. Let A be an n × n matrix. If f is a C -valued function on C , denote by f ( A ) the value of the primary matrix function associated with f on the matrix A. If F is a symmetric C -valued function on C n , let F ( A ) be the value of F on the eigenvalues of A. Generalizing Marcus and Watkins' result we describe those functions f and F, for which [ F ( f ( A j k ) ) ] is positive semidefinite whenever [ A j k ] is positive semidefinite. We extend an example by Choudhury, which gives a negative result for 3 × 3 block matrices.

We prove a Saad's type bound for harmonic Ritz vectors of a Hermitian matrix. The new bound reveals a dependence of the harmonic Rayleigh–Ritz procedure on the condition number of a shifted problem operator. Several practical implications are discussed. In particular, the bound motivates incorporation of preconditioning into the harmonic Rayleigh–Ritz scheme.

The paper is devoted to the structure and the asymptotics of the eigenvector matrix of Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The results extend existing results on banded Toeplitz matrices to full Toeplitz matrices with temperate decay of the entries in the first row and column. We establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors. These formulas are in terms of the Wiener–Hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue. The paper does not aim at providing numerical algorithms. Its main purpose is rather to reveal the structure underneath the eigenvectors, which are shown to be the sum of two harmonics and certain edge corrections.

The First Szegő Limit Theorem gives a remarkable connection between the eigenvalues distribution of a Hermitian Toeplitz matrix and its symbol. This result was extended by Avram and Parter to the singular values of complex Toeplitz matrices. The purpose of this note is to extend their results to a larger class of matrices whose entries are equidistributed and have small mean variation. We also present applications to Kac–Murdock–Szegő matrices, block Toeplitz and locally Toeplitz matrices.

Given an approximating subspace for a Hermitian matrix A , the Rayleigh–Ritz procedure is commonly used to compute a few approximate eigenvalues (called Ritz values) and corresponding approximate eigenvectors (called Ritz vectors). In this paper, new bounds on the canonical angles between the invariant subspace of A associated with its few extreme (smallest or largest) eigenvalues and its approximating Ritz subspace in terms of the differences between Ritz values and the targeted eigenvalues are obtained. From this result, various bounds are readily available to estimate how accurate the Ritz vectors computed from the approximating subspace may be, based on information on approximation accuracies in the Ritz values. The result is helpful in understanding how Ritz vectors move towards eigenvectors while Ritz values are made to move towards eigenvalues.

Motivated by [8] we study the existence of the inverse of an infinite Hermitian positive definite matrix (in short, HPD matrix) from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections. We prove a sufficient condition to assure the inversion of an HPD matrix with square summable rows. For infinite Toeplitz matrices we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of an infinite Toeplitz positive definite matrix is asymptotic Toeplitz. Such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials. We apply these results in the context of the theory of orthogonal polynomials. In particular, we show that for measures on the unit circle T verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures.

For a given real or complex matrix we derive perturbation bounds for some or all singular values. The bounds are of the same quality as corresponding residual bounds for the Hermitian eigenvalue problem. They are suitable for numerical computation.

We present a new algorithm for solving the eigenvalue problem for an n × n real symmetric arrowhead matrix. The algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in O ( n 2 ) operations under certain circumstances. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to Hermitian arrowhead matrices, real symmetric diagonal-plus-rank-one matrices and singular value decomposition of real triangular arrowhead matrices.

How to expand a given subspace efficiently is an important task for large sparse eigenvalue problems. In this paper, we are interested in the following question: which vector from a given subspace, after multiplied by the matrix A , will give a better expanded search subspace. In (2008) [33] , Ye investigated this problem and determined how to choose the optimal expansion vector theoretically. Unfortunately, the result is not computable since it involves some unknown information on the desired eigenvector. When A is symmetric, it was suggested that the Ritz vector may lead to a good candidate for subspace expansion. However, when A is non-Hermitian, the Ritz vector may not be a satisfactory choice. The contribution of this paper is twofold. First, we suggest to use the refined Ritz vector to expand the search subspace, and propose a residual expansion Arnoldi method for subspace expansion. Theoretical results justify the use of the refined Ritz vector. Second, we prove that the elements of the primitive refined Ritz vector have a decreasing pattern going to zero. We then show that the decreasing pattern exists in an arbitrary primitive approximate eigenvector, and derive an inexact residual expansion Arnoldi method for subspace expansion. Numerical examples show the effectiveness of our theoretical results and illustrate the numerical behavior of the new method for subspace expansion.

Given an Hermitian matrix, whose graph is a tree, having a multiple eigenvalue λ , the Parter–Wiener theorem guarantees the existence of principal submatrices for which the multiplicity of λ increases. The vertices of the tree whose removal gives rise to these principal submatrices are called weak Parter vertices and with some additional conditions are called Parter vertices. A set of k Parter vertices whose removal increases the multiplicity of λ by k is called Parter set. As observed by several authors a set of Parter vertices is not necessarily a Parter set. In this paper we prove that if A is a symmetric matrix, whose graph is a tree, and λ is an eigenvalue of A whose multiplicity does not exceed 3, then every set of Parter vertices, for λ relative to A , is also a Parter set.

Let H = ( M K K ⁎ N ) be a positive semidefinite block matrix with square matrices M and N of the same order and denote i = − 1 . The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, λ ( H ) ≺ 1 2 λ ( [ M + N + i ( z K ⁎ − z ¯ K ) ] ⊕ O ) + 1 2 λ ( [ M + N + i ( z ¯ K − z K ⁎ ) ] ⊕ O ) . If, in addition, K is Hermitian, then for any real number r ∈ [ − 2 , 2 ] , λ ( H ) ≺ 1 2 λ ( ( M + N + r K ) ⊕ O ) + 1 2 λ ( ( M + N − r K ) ⊕ O ) , while if K is skew-Hermitian, then for any real number r ∈ [ − 2 , 2 ] , λ ( H ) ≺ 1 2 λ ( ( M + N + r i K ) ⊕ O ) + 1 2 λ ( ( M + N − r i K ) ⊕ O ) , where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz.

Let R be a nonnegative Hermitian matrix. The energy of R , denoted by E ( R ) , is the sum of absolute values of its eigenvalues. We construct an increasing sequence that converges to the Perron root of R . This sequence yields a decreasing sequence of upper bounds for E ( R ) . We then apply this result to the Laplacian energy of trees of order n , namely to the sum of the absolute values of the eigenvalues of the Laplacian matrix, shifted by − 2 ( n − 1 ) / n .

We give a lower bound for the minimum eigenvalue of the Hermitian Toeplitz matrix T n ( | θ | α ) and a corresponding upper bound for the spectral condition number κ 2 ( T n ( | θ | α ) ) . Our main theorem concerns more general cases and establishes a lower bound for the minimum eigenvalue of T n ( f ) and a corresponding upper bound for κ 2 ( T n ( f ) ) , provided the non-negative real-valued symbol f satisfies certain conditions. We discuss some examples of symbols for which these estimates work and we see how the minimax principle can be applied together with our main result in order to obtain estimates of λ min ( T n ( f ) ) and κ 2 ( T n ( f ) ) even in some cases in which the symbol f does not satisfy the conditions of our main theorem. Finally, we provide an extension of the main result to the block Toeplitz case.

Let w(A) be the numerical range of a matrix A ∈ C n × n and a set of points μ 1 , … , μ n - k ∈ w ( A ) that define the spectrum σ ( B ) of a matrix B ∈ C ( n - k ) × ( n - k ) . The problem of imbedding concerns the existence and construction of an isometry V ∈ C n × ( n - k ) such that B = V ∗ AV and is undertaken in this paper. We initially deal with hermitian matrices, for which it is well known that the necessary and sufficient condition for the imbeddability of matrix B in A is their eigenvalue interlacing, and here we present a formulation for the isometry V , when k ⩾ 1 . Moreover, concerning normal matrices, a criterion for imbeddability is established in terms of the real and imaginary parts of their eigenvalues.

This paper investigates the behavior of Ritz values of normal matrices. We apply Ceva’s theorem, a classical geometric result, to understand the geometric relationship between pairs of Ritz values for 3 × 3 normal non-Hermitian matrices, and then analyze the implications for larger matrices. We find that, in the case of normal non-Hermitian matrices, the geometric constraints on the placement of Ritz values provide less freedom than the Cauchy interlacing theorem in the Hermitian case. Using our results we analyze the restarted Arnoldi method with exact shifts applied to a 3 × 3 normal, non-Hermitian matrix.

In [C.R. Johnson, B. Kroschel, M. Omladic, Eigenvalue multiplicities in principal submatrices, Linear Algebra Appl. 390 (2004) 111-120] a result constraining the eigenvalues of principal submatrices of complex Hermitian matrices, based upon matroid theory, was given. Here we give a matricial proof of this result which also enables us to find a generalization of the original result.

This paper is concerned with X H AX ) subject to X H BX = J for a Hermitian matrix pencil A - λ B, where J is diagonal and J 2 = I (the identity matrix of apt size). The same problem was investigated earlier by Kovac-Striko and Veselic ( Linear Algebra Appl. 216 (1995) 139–158) for the case in which B is assumed nonsingular. But in this paper, B is no longer assumed nonsingular, and in fact A - λ B is even allowed to be a singular pencil. It is proved, among others, that the infimum is finite if and only if A - λ B is a positive semi-definite pencil (in the sense that there is a real number λ 0 such that A - λ 0 B is positive semi-definite). The infimum, when finite, can be expressed in terms of the finite eigenvalues of A - λ B . Sufficient and necessary conditions for the attainability of the infimum are also obtained.

The complete relationship between the kth anti-diagonals and the eigenvalues of an n × n ( k ⩽ n ) real symmetric matrix is obtained. When k is even, the relationship is weak majorization. The Hermitian case is also studied. Similar results are obtained for real and complex skew symmetric matrices.

In matrix computations, such as in factoring matrices, Hermitian and, preferably, positive definite elements are occasionally required. Related problems can often be cast as those of existence of respective elements in a matrix subspace. For two dimensional matrix subspaces, first results in this regard are due to Finsler. To assess positive definiteness in larger dimensional cases, the task becomes computational geometric for the joint numerical range in a natural way. The Hermitian element of the Frobenius norm one with the maximal least eigenvalue is found. To this end, extreme eigenvalue computations are combined with ellipsoid and perceptron algorithms.

We consider Hermitian matrix valued functions depending on three parameters that vary in a bounded surface of R 3 . We study how to detect when such functions have coalescing eigenvalues inside this surface. Our criterion to locate these singularities is based on a construction suggested by Stone [20] . For generic coalescings, any such singularity is related to a particular accumulation of a certain phase, or lack thereof, as we cover the surface.

The interlacing theorem of Cauchy–Poincare states that the eigenvalues of a principal submatrix A 0 of a Hermitian matrix A interlace the eigenvalues of A . Fan and Pall obtained an analog of this theorem for normal matrices. In this note we investigate analogs of Cauchy–Poincare and Fan–Pall interlacing theorems for J -Hermitian and J -normal matrices. The corresponding inverse spectral problems are also considered.

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h , is described by the function h in the “Szego way”. Then, using Mergelyan’s theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol h belongs to the Tilli class. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given.

When can an ( n - k ) × ( n - k ) normal matrix B be imbedded in an n × n normal matrix A? This question was studied for the first time 50 years ago by Ky Fan and Gordon Pall, who gave the complete answer in the case k = 1 . Since then, a few authors obtained additional results. In this note, we show how an approach inspired by the Hermitian case can throw some light on the problem.

If k ⩽ n , then G k , n denotes the set of all strictly increasing functions from { 1 , 2 , … , k } to { 1 , 2 , … , n } ordered lexicographically. If A = [ a ij ] is an n × n complex matrix, then A [ α ∣ β ] denotes the k × k submatrix of A whose rows and columns are specified by α and β , respectively, and A ( α ∣ β ) is the corresponding complementary submatrix of A. The k-th permanental compound, CP k ( A ) , of A is the n k × n k matrix indexed by the members of G k , n such that ( CP k ( A ) ) α , β = per ( A [ α ∣ β ] ) for all α , β ∈ G k , n . Much work has been done on permanental compounds. We consider the n k × n k matrix C k ( A ) such that ( C k ( A ) ) α , β = per ( A ( α ∣ β ) ) per ( A [ α ∣ β ] ) . In 1986 Bapat and Sunder [1] conjectured that if A is positive semi-definite and Hermitian, then per ( A ) is the largest eigenvalue C 1 ( A ) . We extend the conjecture to all of the matrices C k ( A ) where 1 ⩽ k ⩽ n . We show how our extended conjecture is related to a theorem of this author, and derive several inequalities. For example, we show that if x is a ( 0 , 1 ) -vector, then 〈 C k ( A ) x , x 〉 ⩽ ‖ x ‖ 2 per ( A ) for all positive semi-definite A.

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.

This paper concerns quadratic matrix functions of the form L(λ)=Mλ2+Dλ+K where M,D,K are Hermitian n×n matrices with M>0. It is shown how new systems of the same type can be generated with some eigenvalues and/or eigenvectors updated and this is accomplished without “spill-over” (i.e. other spectral data remain undisturbed). Furthermore, symmetry is preserved. The methods also apply for Hermitian matrix polynomials of higher degree.

In order to estimate the condition number of the preconditioned matrix proposed in [F.R. Lin, W.K. Ching, Inverse Toeplitz preconditioners for Hermitian Toeplitz systems, Numer. Linear Algebra Appl. 12 (2005) 221–229], we study the inverse of band triangular Toeplitz matrix. We derive an explicit formula for the entries of the inverse of band lower triangular Toeplitz matrix by means of divided difference and use the formula to estimate the condition number of the preconditioned matrices. In particular, we prove that the minimal eigenvalue of preconditioned matrix is well separated from the origin.

For any n -by- n complex matrix A , we use the joint numerical range W ( A , A 2 , … , A k ) to study the polynomial numerical hull of order k of A , denoted by V k ( A ) . We give an analytic description of V 2 ( A ) when A is normal. The result is then used to characterize those normal matrices A satisfying V 2 ( A ) = σ ( A ) , and to show that a unitary matrix A satisfies V 2 ( A ) = σ ( A ) if and only if its eigenvalues lie in a semicircle, where σ ( A ) denotes the spectrum of A . When A = diag ( 1 , w , … , w n - 1 ) with w = e i 2 π / n , we determine V k ( A ) for k ∈ { 2 } ∪ { j ∈ N : j ⩾ n / 2 } . We also consider matrices A ∈ M n such that A 2 is Hermitian. For such matrices we show that V 4 ( A ) is the spectrum of A , and give a description of the set V 2 ( A ) .

Write μ ( A ) = μ 1 ( A ) ⩾ ⋯ ⩾ μ min ( A ) for the eigenvalues of a Hermitian matrix A. Our main result is: Let A be a Hermitian matrix partitioned into r × r blocks so that all diagonal blocks are zero. Then for every real diagonal matrix B of the same size as A μ ( B - A ) ⩾ μ B + 1 r - 1 A . Let G be a nonempty graph, χ ( G ) be its chromatic number, A be its adjacency matrix, and L be its Laplacian. The above inequality implies the well-known result of Hoffman χ ( G ) ⩾ 1 + μ ( A ) - μ min ( A ) , and also, χ ( G ) ⩾ 1 + μ ( A ) μ ( L ) - μ ( A ) . Equality holds in the latter inequality if and only if every two color classes of G induce a ∣ μ min ( A ) ∣ -regular subgraph.

We study group induced cone (GIC) orderings generating normal maps. Examples of normal maps cover, among others, the eigenvalue map on the space of n × n Hermitian matrices as well as the singular value map on n × n complex matrices. In this paper, given two linear spaces equipped with GIC orderings induced by groups of orthogonal operators, we investigate linear operators preserving normal maps of the orderings. A characterization of the preservers is obtained in terms of the groups. The result is applied to show that the normal structure of the spaces is preserved under the action of the operators. In addition, examples are given.

In this short note, it is proved that given any positive definite Hermitian matrix, the eigenvalues of the superoptimal preconditioned matrix do not exceed the corresponding eigenvalues of the optimal preconditioned matrix.