A partial Horn recursion in the cohomology of flag varieties

Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller” flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185–228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written in terms of structure constants coming from induced Grassmannians.     

Estimates for the eigenvalues of the Jordan product of Hermitian matrices

Given Hermitian matrices A and B, Professor Taussky-Todd posed the problem of estimating the eigenvalues of their Jordan product AB+BA. Here we establish bounds for all the eigenvalues of the Jordan product when both A and B are positive definite. At the same time we give a more straightforward proof and an improvement of estimates given by D. W. Nicholson for the smallest eigenvalue.     

Pertubation bounds for the definite generalized eigenvalue problem

Let A and B be Hermitian matrices, and let c(A, B) = inf{|x^H(A + iB)x|:6 = 1}. The eigenvalue problem Ax = λBx is called definite if c(A, B)>0. It is shown that a definite problem has a complete system of eigenvectors and that its eigenvalues are real. Under pertubations of A and B, the eigenvalues behave like the eigenvalues of a Hermitian matrix in the sense that there is a 1-1 pairing of the eigenvalues with the perturbed eigenvalues and a uniform bound for their differences (in this case in the chordal metric). Pertubation bounds are also developed for eigenvectors and eigenspaces.     

Eigenvalue Inequalities and Schubert Calculus

Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A₁,…, A_r ? ¢^n×n with the spectrum of the sum A₁ + … + A_r. These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices.     

Fast enclosure for all eigenvalues in generalized eigenvalue problems

A fast method for enclosing all eigenvalues in generalized eigenvalue problems Ax=λBx is proposed. Firstly a theorem for enclosing all eigenvalues, which is applicable even if A is not Hermitian and/or B is not Hermitian positive definite, is presented. Next a theorem for accelerating the enclosure is presented. The proposed method is established based on these theorems. Numerical examples show the performance and property of the proposed method. As an application of the proposed method, an efficient method for enclosing all eigenvalues in polynomial eigenvalue problems is also sketched.     

Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen–Loève expansion

We describe randomized algorithms for computing the dominant eigenmodes of the generalized Hermitian eigenvalue problem Ax = λBx, with A Hermitian and B Hermitian and positive definite. The algorithms we describe only require forming operations Ax,Bx and B^?1x and avoid forming square roots of B (or operations of the form, B^1/2x or B^?1/2x). We provide a convergence analysis and a posteriori error bounds and derive some new results that provide insight into the accuracy of the eigenvalue calculations. The error analysis shows that the randomized algorithm is most accurate when the generalized singular values of B^?1A decay rapidly. A randomized algorithm for the generalized singular value decomposition is also provided. Finally, we demonstrate the performance of our algorithm on computing an approximation to the Karhunen–Loève expansion, which involves a computationally intensive generalized Hermitian eigenvalue problem with rapidly decaying eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

Absolute and relative Weyl theorems for generalized eigenvalue problems

Weyl-type eigenvalue perturbation theories are derived for Hermitian definite pencils A-λB, in which B is positive definite. The results provide a one-to-one correspondence between the original and perturbed eigenvalues, and give a uniform perturbation bound. We give both absolute and relative perturbation results, defined in the standard Euclidean metric instead of the chordal metric that is often used.     

A note on the Hadamard product of matrices

It is shown that the smallest eigenvalue of the Hadamard product A × B of two positive definite Hermitian matrices is bounded from below by the smallest eigenvalue of AB^T.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Numerical enclosure for each eigenvalue in generalized eigenvalue problem

An algorithm for enclosing all eigenvalues in generalized eigenvalue problem Ax=λBx is proposed. This algorithm is applicable even if A∈C^n×n is not Hermitian and/or B∈C^n×n is not Hermitian positive definite, and supplies nerror bounds while the algorithm previously developed by the author supplies a single error bound. It is proved that the error bounds obtained by the proposed algorithm are equal or smaller than that by the previous algorithm. Computational cost for the proposed algorithm is similar to that for the previous algorithm. Numerical results show the property of the proposed algorithm.     

Some New Bounds for Singular Values and Eigenvalues of Matrix Products

For two Hermitian matrices A and B, at least one of which is positive semidefinite, we give upper and lower bounds for each eigenvalue of AB in terms of the eigenvalues of A and B. For two complex matrices A,B with known singular values, upper and lower bounds are deduced for each singular value of AB.     

Some interlacing results for indefinite Hermitian matrices

A classical theorem of Cauchy states that the eigenvalues of a principal submatrix A₀ of a Hermitian matrix A interlace the eigenvalues of A. We consider the case of a matrix A which is Hermitian with respect to an indefinite inner product.     

Some new bounds on the spectral radius of matrices

A new lower bound on the smallest eigenvalue τ(AB) for the Fan product of two nonsingular M-matrices A and B is given. Meanwhile, we also obtain a new upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B. These bounds improve some results of Huang (2008) [R. Huang, Some inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl. 428 (2008) 1551-1559].     

A note on Stewart's theorem for definite matrix pairs

Let A and B be n×n Hermitian matrices. The matrix pair (A, B) is called definite pair and the corresponding eigenvalue problem βAx = αBx is definite if c(A, B) ≡ inf_{6x6= 1}{|^H(A+iB)x|} > 0. In this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by G.W. Stewart [2]. Moreover, we prove that this upper bound is a projective metric in the set of n × n definite pairs.     

The recursive nature of cominuscule Schubert calculus

The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities.     

Second cohomology groups for Frobenius kernels and related structures

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U) with coefficients in k.     

Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA * = B

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equations AX = B and AXA ^* = B. We first establish a variety of rank and inertia formulas for calculating the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian definite solutions of the matrix equations AX = B and AXA ^* = B, and then use them to characterize many qualities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations and their variations.