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Abstract

If A is a rectangular matrix of rank r, then A may be written as PSQ where P and Q are invertible matrices and s=(IrOOO). This is the rank normal form of the matrix A.

The purpose of this paper is to exhibit some consequences of this representation form.

Abstract

We introduce and study an envelope-type region ɛ(A) in the complex plane that contains the eigenvalues of a given n×n complex matrix A. ɛ(A) is the intersection of an infinite number of regions defined by cubic curves. The notion and method of construction of ɛ(A) extend the notion of the numerical range of A, F(A), which is known to be an intersection of an infinite number of half-planes; as a consequence, ɛ(A) is contained in F(A) and represents an improvement in localizing the spectrum of A.

Abstract

On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p-norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.

Abstract

In this note, we introduce the asymptotic subspace confinement problem, generalizing the usual concept of convergence in discrete-time linear systems. Instead of precise convergence, subspace confinement only requires the convergence of states to a certain subspace of the state space, offering useful flexibility and applicability. We establish a criterion for deciding the asymptotic subspace confinement, drawing upon a general finiteness result for the infinite product of matrices. Our results indicate that the asymptotic subspace confinement problem is algorithmically decidable when an invariant subspace for the set of matrices and some polytope norms are given.

Abstract

In this paper, we introduce the concept of operator AG-preinvex functions and prove some Hermite-Hadamard type inequalities for these functions. As application, we obtain some unitarily invariant norm inequalities for operators.

Abstract

In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.

Abstract

Given a Hermitian matrix M ∈ M3(ℂ) we describe explicitly the real diagonal matrices DM such that ║M + DM║ ≤ ║M + D║ for all real diagonal matrices D ∈ M3(ℂ), where ║ · ║ denotes the operator norm. Moreover, we generalize our techniques to some n × n cases.

Abstract

The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where kλ=kλkλ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ber(A)=supλΩ|A˜(λ)|=supλΩ|Akλ,kλ|. In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then

ber(AX±XA)ber12(A*A+AA*)ber12(X*X+XX*)

and

ber2(A*XB)X2ber(A*A)ber(B*B).

We also prove the multiplicative inequality

ber(AB)ber(A)ber(B)

Abstract

Numerical stability of two main direct methods for solving the symmetric saddle point problem are analyzed. The first one is a generalization of Golub’s method for the augmented system formulation (ASF) and uses the Householder QR decomposition. The second method is supported by the singular value decomposition (SVD). Numerical comparison of some direct methods are given.

Abstract

We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if f : [0;1) → ℝ is a continuous convex function with f(0) ≤ 0, then

for all operators Ci such that (i=1 , ... , n) for some scalar M ≥ 0, where and