ABSTRACT Let A be a square matrix and let Ω be an open set in the plane containing the spectrum of A. We consider the problem of maximizing the operator norm ∥ f ( A ) ∥ amongst all holomorphic functions f from Ω into the closed unit disk. If f 0 is extremal for this problem and if ∥ f 0 ( A ) ∥ > 1 , then it turns out that the matrix f 0 ( A ) has special properties, among them the fact that its principal

ABSTRACT Let S = U S | S | and T = U T | T | be the polar decompositions of S and T in L ( H ) , respectively. In this paper we study solutions of operator equations which have similar real and positive parts with both the similarities implemented by a single invertible operator. Moreover, we investigate various properties of such solutions S and T.

ABSTRACT Let Φ be a unital positive linear map and let A be a positive invertible operator. We prove that there exist partial isometries U and V such that | Φ ( f ( A ) ) Φ ( A ) Φ ( g ( A ) ) | ≤ U ∗ Φ ( f ( A ) A g ( A ) ) U and Φ f ( A ) − r Φ ( A ) r Φ g ( A ) − r ≤ V ∗ Φ f ( A ) − r A r g ( A ) − r V hold under some mild operator convex conditions and some positive numbers r. Further, we show

Let H be a complex Hilbert space, and A be a positive bounded linear operator on H . Let B A ( H ) denote the set of all bounded linear operators on H whose A-adjoint exists. Let A denote a 2 × 2 operator matrix of the form A O O A . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved

ABSTRACT The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this paper, we generalize the commuting variety by using the commuting distance of matrices. We show that over an algebraically closed field, each of our sets

ABSTRACT The purpose of this paper is to establish some new weak majorizations concerning the singular values of the powers of a matrix, which generalize previous related results of Bapat [Majorization and singular values. Linear Multilinear Algebra 1987;21:211–214].

ABSTRACT In this paper, we give the generalization of Cauchy–Binet, Laplace, Sylvester and generalized Dodgson's condensation formulas for the case of rectangular determinants. The proofs are bijective combinatorial proofs similar to that of Zeilberger's paper [Zeilberger D. Dodgson's determinant-evaluation rule proved by TWO-TIMING MEN and WOMEN. Electron J Comb. 1997;4(2):R22; Zeilberger D. A combinatorial

ABSTRACT We aim to give Ando–Hiai-type ratio inequalities for an operator power mean P t ( ω ; A ) and its reverse ones which represent a relation between P t ( ω ; A r ) and P t ( ω ; A ) for r>0 under k-tuple of positive invertible operators A = ( A 1 , … , A k ) and t ∈ ( 0 , 1 ] . As applications, we consider complementary inequalities related to the information monotonicity property via any unital

Let ( M , τ ) be a semi-finite von Neumann algebra, L 0 ( M ) be the set of all τ-measurable operators, μ t ( x ) be the generalized singular number of x ∈ L 0 ( M ) and f : [ 0 , ∞ ) → [ 0 , ∞ ) be a concave function. We proved that if x 1 , x 2 , … , x n are normal operators in L 0 ( M ) , then μ ( f ( | ∑ k = 1 n x k | ) ) is submajorized by μ ( f ( ∑ k = 1 n | x k | ) ) . As an application, we

Motivated by Horn's log-majorization (singular value) inequality s ( A B ) ≺ l o g ⁡ s ( A ) ∗ s ( B ) and the related weak-majorization inequality s ( A B ) ≺ w ⁡ s ( A ) ∗ s ( B ) for square complex matrices, we consider their Hermitian analogs λ ( A B A ) ≺ l o g ⁡ λ ( A ) ∗ λ ( B ) for positive semidefinite matrices and λ ( | A ∘ B | ) ≺ w ⁡ λ ( | A | ) ∗ λ ( | B | ) for general (Hermitian) matrices

ABSTRACT Let ( H ,   ⋅ ,   ∘ ) be a Hopf brace. We first show that the category of left compatible modules over ( H ,   ⋅ ,   ∘ ) is a tensor category. Next, we show that its Drinfeld centre is equivalent to the category H H ′ Y D of left Yetter–Drinfeld modules over ( H ,   ⋅ ,   ∘ ) as a braided tensor category. Finally, we show that Radford's biproduct ( R ♯ H , ⋅ ) and ( R ♯ H ∘ , ∘ ) constitute

ABSTRACT Let p 1 , p 2 , … , p n be distinct positive real numbers and m be any integer. Every symmetric polynomial f ( x , y ) ∈ C [ x , y ] induces a symmetric matrix f ( p i , p j ) i , j = 1 n . We obtain the determinants of such matrices with an aim to find the determinants of P m = ( p i + p j ) m i , j = 1 n and B 2 m = ( p i − p j ) 2 m i , j = 1 n for m ∈ N (where N is the set of natural numbers)

ABSTRACT Let G be a simple undirected connected graph. In this article, we find bounds on the Reciprocal Distance Energy, Reciprocal Distance Laplacian Energy and Reciprocal Distance signless Laplacian Energy. Furthermore, we characterized the graphs that attained some of those bounds.

Let g be the Witt algebra over an infinite field of prime characteristic p. In this paper, we discuss the properties of local derivations on g , and prove that every local derivation on g is a derivation. Moreover, it is shown that there exist some local derivations which are not derivations for the maximal subalgebra g 0 of g .

ABSTRACT We investigate the change in the multiplicities of an eigenvalue of an Hermitian matrix whose graph is a general undirected graph, when an edge is removed from the graph. The same question when the graph is a tree has been investigated in prior work. Here, we give possible classifications of edges in a general graph in terms of the statuses of adjacent vertices. It turns out that there are

Let G be an undirected graph with adjacency matrix A ( G ) . For a vertex u ∈ V ( G ) , the subgraph centrality of u in G is C S ( u ) = ( exp ⁡ ( A ( G ) ) ) u u = ∑ k = 0 ∞ 1 k ! ( A ( G ) k ) u u . In this paper, we give some formulas for C S ( u ) by using equitable partitions and star sets of G.

Suppose that Z is a locally compact Hausdorff space and Ψ , Φ : E ⟶ F are C 0 ( Z ) -module maps between Hilbert C 0 ( Z ) -modules such that for every x , y ∈ E , x ⊥ y implies Ψ ( x ) ⊥ Φ ( y ) . Then there exists a bounded complex-valued function φ on Z that is continuous on Z E = { z ∈ Z : ⟨ x , x ⟩ ( z ) ≠ 0   f o r s o m e   x ∈ E } and satisfies ⟨ Ψ ( x ) , Φ ( y ) ⟩ = φ ⋅ ⟨ x , y ⟩ o n Z ,

In this paper, we study a fast algorithm for the numerical solution of the 1D distributed-order space-fractional diffusion equation. After discretization by the finite difference method, the resulting system is the symmetric positive definite Toeplitz matrix. The preconditioned conjugate gradient method with a circulant preconditioner is employed to solve the linear system. Theoretically, the spectrum

The present paper provides a procedure for constructing full-spark Parseval frames arising from the linear action of a 1-parameter group acting in R n . Precisely, given a square matrix A of order n, with real entries, we say that A induces the full spark frame property if the following conditions hold. There exists a vector v for which given any finite set X ⊂ R of cardinality n, the collection {

We study the eigenvalue problem for the discrete Fourier transform (DFT) and the recently introduced collapsed DFT (CDFT). For the CDFT we in certain cases compute its symmetric rank, show it is not orthogonally decomposable, and compute its eigenvalues and eigenvectors. We generalize the theory of eigenvalues and eigenvectors for symmetric tensors to tensor products of symmetric tensors and apply

Assume that F is an algebraically closed field and let q denote a nonzero scalar in F that is not a root of unity. The universal DAHA (double affine Hecke algebra) H q of type ( C 1 ∨ , C 1 ) is an unital associative F -algebra defined by generators and relations. The generators are { t i ± 1 } i = 0 3 and the relations assert that t i t i − 1 = t i − 1 t i = 1 f o r   a l l   i = 0 , 1 , 2 , 3 ; t

A supertree is a connected and acyclic hypergraph. Let T m △ , △ be the set of the r-uniform supertrees with m edges and exactly two vertices of maximum degree ▵, where m ≥   5 and 3 ≤ △ ≤ [ ( m + 1 ) / 2 ] . We investigate the supertree with the minimum spectral radius among T m △ , △ . Several useful grafting operations are introduced for comparing the spectral radii of supertrees. The supertree

In [A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra. 2010;58(3-4):387–390], Butler constructed a family of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. In this article, we extend this construction for generating larger classes of bipartite graphs, which are cospectral for both the

Let G be a graph with vertex set V ( G ) = { v 1 , … , v n } and E P ( G ) be an n × n matrix whose ( i , j ) -entry is the maximum number of internally edge-disjoint paths between v i and v j , if i ≠ j , and zero otherwise. Also, define E P ¯ ( G ) = E P ( G ) + D , where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing v i , whose E P ( G ) is a

We introduce Ω-sp-algebras as a generalization of dialgebras to the case of Ω-algebras. Given a variety of algebras V , we provide a criterion for an Ω-sp-algebra to be a V -sp-algebra. We give examples of Ω-sp-algebras such as associative sp-systems, ternary Filippov sp-algebras, Lie triple sp-systems, and Bol sp-algebras. We prove a lifting theorem for the term functor and the triviality of every

Let A ∈ C n × n be a normal matrix with spectrum { λ i } i = 1 n , and let A ~ = A + E ∈ C n × n be a perturbed matrix with spectrum { λ ~ i } i = 1 n . If A ~ is still normal, the celebrated Hoffman–Wielandt theorem states that there exists a permutation π of { 1 , … , n } such that ( ∑ i = 1 n | λ ~ π ( i ) − λ i | 2 ) 1 / 2 ≤ ∥ E ∥ F , where ∥ ⋅ ∥ F denotes the Frobenius norm of a matrix. This theorem

In this note, we consider the problem of characterizing the conditions under which the positive semi-definite pth root of a positive semi-definite doubly stochastic matrix is doubly stochastic. First, we obtain new sufficient conditions for this problem that improve the existing ones for the case p = 2. In addition, if we let K n denote the set of all n × n positive semi-definite doubly stochastic

Let R be a ring and Z ( R ) = Z l ( R ) ∪ Z r ( R ) , where Z l ( R ) and Z r ( R ) are the sets of all left and right zero-divisors of R, respectively. The zero-divisor graph of R, denoted by Γ ( R ) , is a simple undirected graph with vertex set Z ∗ ( R ) = Z ( R ) ∖ { 0 } , and two distinct vertices a , b ∈ Z ∗ ( R ) are adjacent if and only if ab = 0 or ba = 0. Let n ≥ 2 , Z n the ring of integers

The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing the derivations of this class of algebras for fields of any characteristic.

We investigate the operators T on a Hilbert space H which have 2-isometric liftings S with the property S ∗ S H ⊂ H . We show that such liftings are closely related to some extensions of T, which have 2 × 2 upper triangular block matrices having contractive entries. We describe these extensions by the corresponding 2-isometric liftings, and respectively as regular A-contractions for some positive operators

Let A be a triangular algebra and σ be an automorphism of A . We consider the problem of describing the form of Lie σ-derivations of A . In particular, we give sufficient conditions that every Lie σ-derivation d of A is the sum d = Δ + γ , where Δ is a σ-derivation of A and γ is a linear mapping from A to its σ-centre that vanishes on A , A . As an application, Lie σ-derivations of (block) upper triangular

Let I n be the n × n identity matrix and J = 0 I n − I n 0 . A matrix A is called symplectic if A T J A = J . A symplectic matrix A is a commutator of symplectic involutions if A = X Y X − 1 Y − 1 , where X and Y are symplectic matrices satisfying X 2 = Y 2 = I . In this article, we give necessary and sufficient condition for a symplectic matrix over the complex number field to be expressed as a product

Based on the non-alternating preconditioned HSS iteration scheme, a class of the Uzawa-NPHSS iteration method for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite is established. The convergence properties for the nonsingular saddle point problems and the semi-convergence properties for the singular

The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in the literature. In this article, we present many new inequalities for the numerical radius of accretive matrices. The importance of this study is the presence of a

For a simple connected graph G, let D ( G ) , T r ( G ) , D L ( G ) and D Q ( G ) , respectively, are the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix. The generalized distance matrix D α ( G ) of G is the convex linear combinations of T r ( G ) and D ( G ) and is defined as D α ( G ) = α T r ( G ) + ( 1 − α )

We consider algebraic systems with several products, including unary products, S (as examples we can take linear spaces, algebras, superalgebras, hom-algebras, triple systems, hom-triple systems, Poisson algebras, Bol algebras, n-algebras, etc.) We show that any basis of S gives rise to a decomposition of S as a direct sum of indecomposable well-described ideals (fine decomposition). The simplicity

A mathematical model is identifiable if its parameters can be recovered from data. Here we investigate, for linear compartmental models, whether (local, generic) identifiability is preserved when parts of the model – specifically, inputs, outputs, leaks, and edges – are moved, added, or deleted. Our results are as follows. First, for certain catenary, cycle, and mammillary models, moving or deleting

Let G be a graph having a unique perfect matching M, A ( G ) be the adjacency matrix of G and W ( G ) be the collection of all positive weight functions defined on the edge set of G in which each weight function w assigns weight 1 to each matching edge and a positive weight to each non-matching edge. The weighted graph G w satisfies the ( − S R ) property if for each eigenvalue of G w , its anti-reciprocal

The class of non-bipartite unicyclic graphs with a unique perfect matching, denoted by U , is considered in this article. This article describes the entries of the inverse of the adjacency matrix of a graph in U . It is proved that the inverse graph of a graph in U is always non-bipartite. In this article, we characterize the graphs in U whose inverses are mixed graphs. Among all such graphs in U we

The classical completion problem of describing the possible similarity class of a square matrix with a prescribed arbitrary submatrix was studied by many authors through time, and it is completely solved in [Dodig M, Stošić M. Similarity class of a matrix with prescribed submatrix. Linear Multilinear Algebra. 2009;57:217–245; Combinatorics of polynomial chains. Linear Algebra Appl. 2020;589:130–157]

The aim of the paper is to give a description of nonlinear Lie centralizers for a certain class of generalized matrix algebras. The result is then applied to some full matrix algebras and triangular algebras.

The purpose of this short note is to consider multivariate Hasse-Schmidt derivations on exterior algebras and to show how they easily provide remarkable identities, holding in the algebra of square matrices, which generalize the classical theorem of Cayley–Hamilton.

New numerical radius inequalities for Hilbert space operators are given. These inequalities improve a well-known inequality of F. Kittaneh. We also show that the inequalities obtained here are sharp. A refinement of the triangle inequality is also shown.

Let H be a complex Hilbert space, and A be a positive bounded linear operator on H . Let B A ( H ) denote the set of all bounded linear operators on H whose A-adjoint exists. Let A denote a 2 × 2 operator matrix of the form A O O A . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved

A linear map ψ on a Lie algebra g over a field F with char ( F ) ≠ 2 is called to be commuting (resp., skew-commuting) if [ ψ ( x ) , y ] = [ x , ψ ( y ) ] (resp., [ ψ ( x ) , y ] = − [ x , ψ ( y ) ] ) for all x , y ∈ g , and to be strong commutativity-preserving if [ ψ ( x ) , ψ ( y ) ] = [ x , y ] for all x , y ∈ g . Let L be a finite-dimensional simple Lie algebra over an algebraically closed field

A Pascal-like matrix is constructed whose row entries are the terms of the modified Hermite polynomials of two variables. The multiplication of the two Pascal-like matrices and the power and inverse of the Pascal-like matrix have very similar results to the well-known Binomial Matrix, and the factorization of the Pascal-like matrix has also been given, which is completely similar to the factorization

We investigate some aspects of various numerical radius orthogonalities and numerical radius parallelism for bounded linear operators on a Hilbert space H . Among several results, we show that if T , S ∈ B ( H ) and M ω ( T ) ∗ = M ω ( S ) ∗ , then T ⊥ ω B S if and only if S ⊥ ω B T , where M ω ( T ) ∗ = { { x n } : ∥ x n ∥ = 1 , lim n | ⟨ T x n , x n ⟩ | = ω ( T ) } , and ω ( T ) is the numerical

In this paper, we consider the quaternion matrix equation X − A X ˆ B = C , and study its minimal norm least squares solution, j-self-conjugate least-squares solution and anti-j-self-conjugate least-squares solution. By the real representation matrices of quaternion matrices, their particular structure and the properties of Frobenius norm, we convert above least-squares problems into corresponding

We present several new matrix inequalities which connect the Schur complement, sector matrices and weighted arithmetic mean, geometric mean and harmonic mean, extending a key result in Lin [Extension of a result of Hanynsworth and Hartfiel. Arch Math. 2015;1:93–100].

In this paper, we continue the study of abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable and nilpotent Lie algebras. We show that supersolvable Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two, and that the same is true for nilpotent Lie

In this paper, we review and investigate some properties of a linear pencil in the frame of the point spectrum and an isolated point in its spectrum. We next give relationships among the spectrum of the linear pencil, the Taylor spectrums of a commuting pair and its spherical Aluthge transform through projection maps and the spectral mapping theorem.

Some sufficient conditions ensuring that the subdirect sum of p-norm strictly diagonally dominant [or, for short, SDD(p)] matrices is in the class of SDD(p) matrices, are given. In particular, it is shown that the subdirect sum of overlapping principal submatrices of an SDD(p) matrix is also an SDD(p) matrix. Some examples are also given that illustrate the conditions presented.

This article considers some fundamental estimation problems on a general partitioned linear model with partial parameter restrictions. We shall derive analytical formulas for calculating the ordinary least-squares estimators (OLSEs) and the best linear unbiased estimators (BLUEs) of the whole and partial parameter vectors in the model and its reduced models, and establish identifying conditions for

We study BDF theory in the context of (not necessarily simple) purely infinite corona algebras. Let B be a nonunital separable simple C*-algebra with standard regularity properties for which C ( B ) is purely infinite (though not necessarily simple). Let A be a separable nuclear C*-algebra. We prove a BDF Voiculescu decomposition theorem for maps from A to C ( B ) , and use this to prove that E x t

The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many of the commonly studied spectra of graphs. We show that for a large class of graphs these eigenvalues can be computed explicitly. We also present the applications

Let X be an infinite-dimensional Banach space and let B ( X ) denote the algebra of all bounded linear operators on X. For T ∈ B ( X ) the sets σ 1 ( T ) = { λ ∈ σ ( T ) ∣ T − λ I  is a semi-Fredholm operator } , σ 2 ( T ) = { λ ∈ σ ( T ) ∣ T − λ I  is a Fredholm operator } , and σ 3 ( T ) = { λ ∈ σ ( T ) ∣ T − λ I  is a Weyl operator } are called, respectively, the semi-Fredholm domain, the Fredholm

Let G be a graph and S ( G ) be the Seidel matrix of G. The Seidel energy E s ( G ) is defined to be the sum of the absolute value of all eigenvalues of the Seidel matrix of G. In this paper, we study some properties of the Seidel eigenvalues of G. Moreover, we present some lower and upper bounds for the Seidel energy of G.

In this paper, we investigate the 2-local isometries between the vector-valued differentiable function spaces C 0 p ( X , E ) and C 0 q ( Y , E ) , where X, Y are open subsets of R and E is a smooth, strictly convex, reflexive and 2-iso-reflexive Banach space (in particular, when 1 ≤ p < ∞ and p ≠   2 , ℓ p satisfies such conditions). We show that such maps can be written as weighted composition operators

Let R be a prime algebra over a commutative ring K, let I be a nonzero ideal of R and let f ( X 1 , … , X k ) be a multilinear polynomial over K in k non-commuting indeterminates which is not central-valued on R. Suppose that a ∈ R and d is a derivation of R such that a ( d ( f ( x 1 , … , x k ) ) s − γ f ( x 1 , … , x k ) t ) m ∈ Z ( R ) for all x 1 , … , x k ∈ I , where γ ∈ K , s, t, m are fixed

We give an intriguing construction for establishing a sequence of unital, commutative, associative number systems by introducing the Z 2 -graded algebra S L n ( R ) . We show how the algebra S L n ( R ) arises naturally from the Pauli matrices. Moreover, we investigate eigenvalues of elements in S L n ( R ) and the matrix algebra M m ( S L n ( R ) ) .