Linear Algebra and its Applications

article

A unified approach to operator monotone functions

Tianpei Jiang, Hristo Sendov

Linear Algebra and its Applications > 2018 > 541 > C > 185-210

The notion of operator monotonicity dates back to a work by Löwner in 1934. A map F:Sn→Sm is called operator monotone, if A⪰B implies F(A)⪰F(B). (Here, Sn is the space of symmetric matrices with the semidefinite partial order ⪰.) Often, the function F is defined in terms of an underlying simpler function f. Of main interest is to find the properties of f that characterize operator monotonicity of...

article

The characteristic polynomial of lexicographic product of graphs

Zhijun Wang, Dein Wong

Linear Algebra and its Applications > 2018 > 541 > C > 177-184

Let G and H be two simple graphs. In this paper, the characteristic polynomial of G[H], the lexicographic product of G and H, is determined. As an application, the characteristic polynomial and the spectrum of G[H] are obtained explicitly when H has exactly one or two main eigenvalues.

article

Improved random perturbation intervals of symmetric eigenvalue problem

Chuanfu Xiao, Hanyu Li

Linear Algebra and its Applications > 2018 > 541 > C > 1-12

In this paper, by transforming the symmetric random perturbed matrix into a special form and using the result from Li and Li (2005) [7], we first present an improved random perturbation interval. Under the random perturbation in the interval, the simple eigenvalue of the original matrix is still simple. Then, we extend the result to the case of several eigenvalues. Finally, numerical experiments are...

article

Horizontal and vertical formulas for exponential Riordan matrices and their applications

Gi-Sang Cheon, Ji-Hwan Jung, Paul Barry

Linear Algebra and its Applications > 2018 > 541 > C > 266-284

In this paper, we show that an infinite lower triangular matrix A=[aij]i,j∈N0 is an exponential Riordan matrix A=E(g,f) given by ∑i≥jaijzi/i!=gfj/j! if and only if there exist both a horizontal pair {hn;h˜n}n≥0 and a vertical pair {vn;v˜n}n≥0 of sequences that represent all the elements in the matrix. As a consequence, we obtain that if the horizontal and vertical pairs of an exponential Riordan matrix...

article

On isomorphisms of pseudo-Euclidean spaces with signature (p,n−p) for p=2,3

N.G. Pavlova, A.O. Remizov

Linear Algebra and its Applications > 2018 > 541 > C > 60-80

As is well known, for every orthogonal transformation of the Euclidean space there exists an orthogonal basis such that the matrix of the transformation is block-diagonal with first order blocks ±1 and second order blocks that are rotations of the Euclidean plane. There exists a natural generalization of this theorem for Lorentz transformations of pseudo-Euclidean spaces with signature (1,n−1). In...

article

On isometry groups of self-adjoint traceless and skew-symmetric matrices

Marcell Gaál, Robert M. Guralnick

Linear Algebra and its Applications > 2018 > 541 > C > 211-220

This note is concerned with isometries on the spaces of self-adjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten p-norm isometries. Furthermore, we point out that our proof techniques could be applied to obtain an old result concerning isometries on skew-symmetric matrices.

article

Editorial Board

Linear Algebra and its Applications > 2018 > 541 > C > ii

article

Regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint

Yan Liu, Yuming Shi

Linear Algebra and its Applications > 2018 > 541 > C > 94-130

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of induced regular SSEs, generated by the same difference expression on smaller finite intervals. It is shown...

article

Multiplicities of distance Laplacian eigenvalues and forbidden subgraphs

Rosário Fernandes, Maria Aguieiras A. de Freitas, Celso M. da Silva, Renata R. Del-Vecchio

Linear Algebra and its Applications > 2018 > 541 > C > 81-93

In this work, the graphs of order n having the second distance Laplacian eigenvalue of multiplicity n−2 are determined. Besides that, this result also characterizes the graphs where the multiplicity of some distance Laplacian eigenvalue is equal to n−2. In addition, all connected graphs of order n where the largest eigenvalue of the distance Laplacian matrix has multiplicity n−3 are determined. Finally,...

article

The noncommutative Löwner theorem for matrix monotone functions over operator systems

J.E. Pascoe

Linear Algebra and its Applications > 2018 > 541 > C > 54-59

Given a function f:(a,b)→R, Löwner's theorem states f is monotone when extended to self-adjoint matrices via the functional calculus, if and only if f extends to a self-map of the complex upper half plane. In recent years, several generalizations of Löwner's theorem have been proven in several variables. We use the relaxed Agler, M^cCarthy, and Young theorem on locally matrix monotone functions in...

article

Noncrossing partitions, noncrossing graphs, and q-permanental equations

Eduardo Marques de Sá

Linear Algebra and its Applications > 2018 > 541 > C > 36-53

We first prove a few simple results to illustrate some algebraic combinatorial features of the q-permanent. This is followed by a characterization of a noncrossing permutation in terms of the numbers of inversions of its cycles. Then we use a family of derivative formulas for the q-permanent of a square matrix A to characterize several structures of noncrossing kind. Each such formula f characterizes...

article

A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth

Jicheng Li, Liqiang Dong, Guo Li

Linear Algebra and its Applications > 2018 > 541 > C > 131-162

In this paper, we propose and discuss a class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth. For an odd 2p+1 with a positive integer p, the problem is to construct an n×n real symmetric banded matrix with bandwidth 2p+1 whose m×m leading principal submatrix is a given m×m real symmetric banded matrix with bandwidth 2p+1 and spectrum is a given set of real numbers...

article

Infinitely divisible nonnegative matrices, M-matrices, and the embedding problem for finite state stationary Markov chains

Alexander Van-Brunt

Linear Algebra and its Applications > 2018 > 541 > C > 163-176

This paper explicitly details the relation between M-matrices, nonnegative roots of nonnegative matrices, and the embedding problem for finite-state stationary Markov chains. The set of nonsingular nonnegative matrices with arbitrary nonnegative roots is shown to be the closure of the set of matrices with matrix roots in IM. The methods presented here employ nothing beyond basic matrix analysis, however...

article

Corrigendum to “SPN graphs: When copositive = SPN” [Linear Algebra Appl. 509 (2016) 82–113]

Naomi Shaked-Monderer

Linear Algebra and its Applications > 2018 > 541 > C > 285-286

In this corrigendum, an error in the proof of Theorem 6.4 in Shaked-Monderer (2016) [2] is pointed out. This theorem states that every graph Tn consisting of n−2 triangles sharing a common base is SPN. We have an alternative proof for the case n=5, but for all n>5 it remains open whether Tn is SPN. As a result, the question whether K2,n, n>4, is SPN also remains open.

article

Note on an upper bound for sum of the Laplacian eigenvalues of a graph

Xiaodan Chen, Jingjian Li, Yingmei Fan

Linear Algebra and its Applications > 2018 > 541 > C > 258-265

For a simple graph G with n vertices and m edges having Laplacian eigenvalues μ1(G)≥μ2(G)≥⋯≥μn(G), let Sk(G) be the sum of k largest Laplacian eigenvalues of G. In this note, we prove that if G is a connected graph of order n≥2 with m edges having clique number ω and vertex covering number τ, thenSk(G)≤k(τ+1)+m−ω(ω−1)2, with equality if k≤ω−1 and G is the graph obtained by joining n−ω pendant vertices...

article

Editorial Board

Linear Algebra and its Applications > 2018 > 541 > C > i

article

Local automorphisms of finitary incidence algebras

Jordan Courtemanche, Manfred Dugas, Daniel Herden

Linear Algebra and its Applications > 2018 > 541 > C > 221-257

Let R be a commutative, indecomposable ring with identity and (P,≤) a partially ordered set. Let FI(P) denote the finitary incidence algebra of (P,≤) over R. We will show that, in most cases, local automorphisms of FI(P) are actually R-algebra automorphisms. In fact, the existence of local automorphisms which fail to be R-algebra automorphisms will depend on the chosen model of set theory and will...

article

Unitary similarity of the determinantal representation of unitary bordering matrices

Mao-Ting Chien, Hiroshi Nakazato

Linear Algebra and its Applications > 2018 > 541 > C > 13-35

Let F(x,y,z) be a hyperbolic ternary form of degree n. The Helton–Vinnikov theorem asserts that there exists an n×n complex symmetric matrix S such that FS(x,y,z)=F(x,y,z), where the determinantal hyperbolic ternary form of an n×n matrix B is defined by FB(x,y,z)=det⁡(xℜ(B)+yℑ(B)+zIn). Let A be an n×n unitary bordering matrix. It is known that A is unitarily similar to a symmetric matrix. In this...

INFONA - science communication portal

Linear Algebra and its Applications

A unified approach to operator monotone functions

The characteristic polynomial of lexicographic product of graphs

Improved random perturbation intervals of symmetric eigenvalue problem

Horizontal and vertical formulas for exponential Riordan matrices and their applications

On isomorphisms of pseudo-Euclidean spaces with signature (p,n−p) for p=2,3

On isometry groups of self-adjoint traceless and skew-symmetric matrices

Editorial Board

Regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint

Multiplicities of distance Laplacian eigenvalues and forbidden subgraphs

The noncommutative Löwner theorem for matrix monotone functions over operator systems

Noncrossing partitions, noncrossing graphs, and q-permanental equations

A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth

Infinitely divisible nonnegative matrices, M-matrices, and the embedding problem for finite state stationary Markov chains

Corrigendum to “SPN graphs: When copositive = SPN” [Linear Algebra Appl. 509 (2016) 82–113]

Note on an upper bound for sum of the Laplacian eigenvalues of a graph

Editorial Board

Local automorphisms of finitary incidence algebras

Unitary similarity of the determinantal representation of unitary bordering matrices

Filter options

Publication date

Keywords

INFONA - science communication portal

Linear Algebra and its Applications

Add recipient

Sending message cancelled

Are you sure you want to cancel sending this message?

Send message

Filter options

Publication date

Date range setting

Set the date range to filter the displayed results. You can set a starting date, ending date or both. You can enter the dates manually or choose them from the calendar.

Keywords

Reporting an error / abuse

Sending the report failed

Accessibility options