Some interlacing theorems on the schur complement

We give a short proof of the interlacing theorem on the eigenvalues of Schur complements of Hermitian matrices due to Hu and Smith, and extend the technique to prove an interlacing theorem on the singular values of Schur complements of rectangular complex matrices.     

A New Proof of the Protasov-Voynov Theorem on Semigroups of Nonnegative Matrices

A new combinatorial proof of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices is proposed. The original proof was obtained by geometric methods.

     

A note on a theorem of minc on irreducible non-negative matrices

A short proof is given for Mine's partial converse to a fundamental theorem of Frobenius on irreducible nonnegative matrices.     

On a theorem of Wigner on products of positive matrices

A simple algebraic proof of a theorem due to Wigner on the product of three positive matrices is given. It is shown that the theorem holds for four matrices under an additional condition. The proofs are valid in the more general case of operators in a Hilbert space.     

Combinatorially symmetric matrices

This paper presents a brief survey of some of the recent results on combinatorially symmetric matrices. Some examples of current interest are given. We present a simple proof of a theorem of Parker and Youngs. We also prove a theorem on skew—symmetrization of combinatorially symmetric matrices and present some applications to stability problems.     

关于稳定矩阵分解定理的一个简单证明

一个矩阵称为稳定的,如果这个矩阵的特征值全包含在单位开圆盘内.利用Parker关于复方阵的分解定理给出了稳定矩阵分解定理的一个简单证明,并对奇异值全部严格小于1的矩阵给出了类似的结论.     

A short proof of Hua's fundamental theorem of the geometry of hermitian matrices

We present a new and simple proof of Hua's fundamental theorem of the geometry of hermitian matrices which characterizes bijective maps preserving adjacency in both directions on the real vector space of all n × n hermitian matrices.     

Analytic proofs of a network feasibility theorem and a theorem of Fulkerson

We provide an elementary proff of Fulkerson's theorem which gives the permutation matrices as extreme points of a certain unbounded convex polyhedron. An adaptation of the proof also establishes an analogous feasibility theorem for network flows which has Fulkerson's theorem as a corollary.     

A short proof of the Arendt-Chernoff-Kato theorem

We give a short new proof of the Arendt-Chernoff-Kato theorem, which characterizes generators of positive C ₀ semigroups in order unit spaces. The proof avoids half-norms and subdifferentials and is based on a sufficient condition for an operator to have positive inverse, which is new even for matrices.     

A proof of Devadze’s theorem on generators of the semigroup of Boolean matrices

In 1968 Devadze described, without a proof, minimal sets of generators of the semigroup of n×n Boolean matrices. We provide a proof of Devadze’s theorem.     

A note on a theorem of minc on irreducible non-negative matrices

A short proof is given for Mine's partial converse to a fundamental theorem of Frobenius on irreducible nonnegative matrices.     

A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

We use ergodic theory to prove a quantitative version of a theorem of M.A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded.     

关于Sylvester惯性律的一种变型

借助连续性推证与Hermite矩阵特征值的Courant-Fischer定理给出关于Sylvester惯性律一种变型的简化证明.     

Affine semigroup rings that are complete intersections

This paper presents a result concerning the structure of affine semigroup rings that are complete intersections. It generalizes to arbitrary dimensions earlier results for semigroups of dimension less than four. The proof depends on a decomposition theorem for mixed dominating matrices.

     

Toeplitz Minors

We give a new proof of the strong Szegö limit theorem estimating the determinants of Toeplitz matrices using symmetric function theory. We also obtain asymptotics for Toeplitz minors.     

Two applications of a consistency theorem for systems of linear inequalities

A consistency criterion for systems of linear inequalities is applied to prove an existence theorem for non-negative matrices with restricted row sums, column sums and diagonal elements. As another application of the consistency criterion, a surprisingly short proof of a theorem of Hardy, Littlewood and Pólya is given.     

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.     

UNCONDITIONAL CAUCHY SERIES AND UNIFORM CONVERGENCE ON MATRICES

The authors obtain new characterizations of unconditional Cauchy series in termsof separation properties of subfamilies of P(N), and a generalization of the Orlicz-PettisTheorem is also obtained. New results on the uniform convergence on matrices anda new version of the Hahn-Schur summation theorem are proved. For matrices whoserows define unconditional Cauchy series, a better sufficient condition for the basicMatrix Theorem of Antosik and Swartz, new necessary conditions and a new proof ofthat theorem are given.     

Linear operators preserving the decomposable numerical radius

The characterization of all linear operators on matrices which preserve the decomposable numerical radius is obtained. This result refines those of Tam. Marcus and Filippenko on the topic. The proof of the main theorem depends on a characterization of scalar multiples of unitary matrices in terms of decomposable numerical radius that is of independent interest.     

Einschliessung von Eigenwerten und Mohrsches Spannungsdiagramm

Summary A new proof of a theorem byWielandt for inclusion of eigenvalues of Hermitian matrices is given. Relations between that inclusion method and the Mohr stress diagram are investigated.