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Let A and B be $ n\times n $ n × n positive semidefinite matrices, $ f:[0,\infty)\rightarrow \lbrack 0,\infty) $ f : [ 0 , ∞) → [ 0 , ∞) be an increasing convex function with $ f\left (0\right) =0 $ f (0) = 0 , and let $ \Phi : \mathbb {M}_{n}(\mathbb {C})\rightarrow \mathbb {M}_{n}(\mathbb {C}) $ Φ : M n (C) → M n (C) be a positive unital linear map. It is shown that $$\begin{align*} & \prod_{j=1}^{k} s_{j}\left(f^{4}\left(\Phi \left(\frac{A+B}{2}\right) \right) \right)\\ & \quad \leq 2^{-4} \prod_{j=1}^{k} \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(A\right) \right) +\Phi \left(f^{2}\left(A\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(B\right) \right) \right\Vert +1\right) \\ & \quad\quad \times \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(B\right) \right) +\Phi \left(f^{2}\left(B\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(A\right) \right) \right\Vert +1\right) \end{align*}$$ ∏ j = 1 k s j (f 4 (Φ (A + B 2))) ≤ 2 − 4 ∏ j = 1 k (s j (Φ 2 (f 2 (A)) + Φ (f 2 (A))) + ‖ Φ (f 2 (B)) ‖ + 1) × (s j (Φ 2 (f 2 (B)) + Φ (f 2 (B))) + ‖ Φ (f 2 (A)) ‖ + 1) for $ k=1,2,\ldots,n $ k = 1 , 2 , ... , n , where $ s_{j}\left (T\right) $ s j (T) and $ \left \Vert T\right \Vert $ ‖ T ‖ are the $ j{\rm th} $ j th singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by Lemos and Soares in 2018. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by Bhatia, Lim and Yamazaki in 2016, and recently proved by Dinh, Dumitru and Franco in 2017 for the Schatten p-norm with 1 ≤ p ≤ ∞. [ABSTRACT FROM AUTHOR]

We present a characterization of eigenvalue inequalities between two Hermitian matrices by means of inertia indices. As applications, we deal with some classical eigenvalue inequalities for Hermitian matrices, including the Cauchy interlacing theorem and the Weyl inequality, in a simple and unified approach. We also give a common generalization of eigenvalue inequalities for (Hermitian) normalized Laplacian matrices of simple (signed, weighted, directed) graphs. Our approach is also suitable for Hermitian matrices of the second kind of digraphs recently introduced by Mohar. [ABSTRACT FROM AUTHOR]

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 four cases that do occur in a general graph but cannot occur in a tree. [ABSTRACT FROM AUTHOR]

Let Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i be a linear map on the Kronecker product of spaces of Hermitian matrices H n i of size n i ≥ 3. (If d=1, we identify ⨂ i = 1 d H n i with H n 1 .) We establish a condition under which Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i if and only if det Ψ ⨂ i = 1 d A i = det ⨂ i = 1 d A i for all ⨂ i = 1 d A i ∈ ⨂ i = 1 d H n i . Then for d ∈ { 1 , 2 } , we apply this fact to characterize maps Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i such that Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i ⨂ i = 1 d with some mild conditions. [ABSTRACT FROM AUTHOR]

The signed enhanced principal rank characteristic sequence (sepr-sequence) of an Hermitian matrix is the sequence , where is either , , , , , , or , based on the following criteria: if B has both a positive and a negative order-k principal minor, and each order-k principal minor is nonzero. (respectively, ) if each order-k principal minor is positive (respectively, negative). if each order-k principal minor is zero. if B has each a positive, a negative and a zero order-k principal minor. (respectively, ) if B has both a zero and a nonzero order-k principal minor, and each nonzero order-k principal minor is positive (respectively, negative). Such sequences provide more information than the epr-sequence in the literature, where the kth term is either , , or based on whether all, none, or some (but not all) of the order-k principal minors of the matrix are nonzero. Various sepr-sequences are shown to be unattainable by Hermitian matrices. In particular, by applying Muir’s law of extensible minors, it is shown that subsequences such as and are prohibited in the sepr-sequence of a Hermitian matrix. For Hermitian matrices of orders , all attainable sepr-sequences are classified. For real symmetric matrices, a complete characterization of the attainable sepr-sequences whose underlying epr-sequence contains as a non-terminal subsequence is established. [ABSTRACT FROM AUTHOR]

In this note, based on the previous work by Pour and Goughery (New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems Numer. Algor. 69 (2015) 207–225), we further discuss this new Hermitian and skew-Hermitian splitting (described as NHSS) methods for non-Hermitian positive definite linear systems. Some new convergence conditions of the NHSS method are obtained, which are superior to the results in the above paper. [ABSTRACT FROM AUTHOR]

In this paper, the solvability conditions and the explicit expressions of the Hermitian solutions to the system of matrix equations and the Hermitian nonnegative definite solutions to the system of matrix equations are, respectively, put forward, by making full use of the generalized inverse and the rank of matrices. As applications, some special cases of the above systems of matrix equations are considered. In addition, the maximal ranks and inertias of the Hermitian solutions are, respectively, presented. [ABSTRACT FROM PUBLISHER]

We characterize all Jordan triple product homomorphisms, that is, mappingssatisfying from the set of all Hermitiancomplex matrices to the field of complex numbers. Further, we characterize all Jordan triple product homomorphisms from the field of complex or real numbers or the set of all nonnegative real numbers to the set of all Hermitiancomplex matrices. [ABSTRACT FROM AUTHOR]

In this note, we characterize linear maps between spaces of Hermitian matrices over a field with involution which preserve the rank of the identity matrix and the rank of tensor products of rank one Hermitian matrices. It is assumed that the fixed field with respect to the field with involution has at least four elements. We also present some applications of our main result to other preserver problems. [ABSTRACT FROM AUTHOR]

Using the method of seeds and branch duplication, it is shown that for every tree of diameter, there is an Hermitian matrix with as few asdistinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter, it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of. Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made. [ABSTRACT FROM PUBLISHER]

A method for the computation ofinverses of a given matrixis derived, based on the full-rankdecomposition of an appropriate matrix. As a corollary, a new method considering the advantages of full-rankdecomposition is developed. It is then specialized to the set of polynomial matrices. Therefore, an algorithm for efficient symbolic computation ofinverses of a polynomial matrix is proposed. An additional diagonal matrixyields to avoiding the computation of entries containing square roots of polynomials, therefore increasing the algorithm’s performances. Some implementation details and comparative processing times to other similar methods are provided, illustrating the algorithm’s efficiency. [ABSTRACT FROM AUTHOR]

Letbe two Hermitian matrices. We propose new perturbation bounds on the differences between the eigenvalues ofandby the bounds of the eigenvector components. These results extend those of Nakatsukasa. [ABSTRACT FROM AUTHOR]

For a positive integer, letbe the set ofcomplex matrices. Supposeare positive integers and. Denote bythe-numerical range of a matrix. It is shown that a linear mapsatisfiesfor allandif and only if there is a unitarysuch that one of the following holds.   For all , ,     and for all , ,where (1)is the identity mapor the transposition map, or (2)andhas the formor. [ABSTRACT FROM PUBLISHER]

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ onmn × mnHermitian matrices such that φ(A ⊗ B) andA ⊗ Bhave the same spectrum for anym × mHermitianAandn × nHermitianB. Such a map has the formA ⊗ B ↦ U(ϕ1(A) ⊗ ϕ2(B))U* formn × mnHermitian matrices in tensor formA ⊗ B, whereUis a unitary matrix, and forj ∈ {1, 2}, ϕjis the identity map X ↦ Xor the transposition map X ↦ Xt. The structure of linear maps leaving invariant the spectral radius of matrices in tensor formA ⊗ Bis also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems. [ABSTRACT FROM AUTHOR]

A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2 × 2 and 3 × 3, as well as k × k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold. [ABSTRACT FROM PUBLISHER]

We study the maps on complex Hermitian and complex symmetric matrices which preserve zeros of a Jordan product. [ABSTRACT FROM AUTHOR]

A bordered matrix is a two-by-two partitioned matrix with its lower-right corner equal to a null matrix. In this article, we present eight partitioned matrices consisting of the Moore-Penrose inverses of submatrices in a bordered matrix, and give necessary and sufficient conditions for the eight partitioned matrices to be generalized inverses of the bordered matrix. [ABSTRACT FROM AUTHOR]

Suppose [image omitted] is a field with proper involution and of arbitrary characteristic. Additive maps, which do not increase rank-one on hermitian matrices with entries from [image omitted] are classified. The result is then used to classify additive maps that do not increase minimal rank on the set of symmetric elements, relative to general involution on a matrix algebra. [ABSTRACT FROM AUTHOR]

Let A =( a ij ) be an n -by- n matrix. For any real µ, define the polynomial where l (s) is the number of inversions of the permutation s in the symmetric group Sn . We prove that P µ (A) is a strictly increasing function of µ ? [-1,1], for a Hermitian positive definite nondiagonal matrix A , whose graph is a tree. [ABSTRACT FROM AUTHOR]