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We classify maximal digraphs whose Hermitian spectral radius is at most 2. [ABSTRACT FROM AUTHOR]

Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon's conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods. [ABSTRACT FROM AUTHOR]

Yoopyo Hong proved in 1989 that each Hermitian matrix A is orthogonally *congruent to a matrix of the form ε 1 A 1 ⊕ ⋯ ε r A r ⊕ B 1 ⊕ ⋯ ⊕ B s , in which A 1 , ... , A r , B 1 , ... , B s are uniquely determined by the orthogonal *congruence class of A, and ε 1 , ... , ε r ∈ { 1 , − 1 }. We prove that ε 1 , ... , ε p are uniquely determined by the orthogonal *congruence class of A as well. As an application, we present a canonical form of a pair (A , B) consisting of a Hermitian matrix A and a nonsingular symmetric matrix B with respect to transformations (S ⁎ A S , S T B S) with a nonsingular S. [ABSTRACT FROM AUTHOR]

Tree tensor networks such as the tensor train (TT) format are a common tool for high-dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While the behavior of such is as essential as in the matrix case, here questions about the feasibility of specific constellations arise: which prescribed tuples can be realized as singular values of a tensor, and what is this feasible set? We first show the equivalence of the tensor feasibility problem (TFP) to the quantum marginal problem (QMP). In higher dimensions, in case of the TT-format, the conditions for feasibility can be decoupled. By present results for three dimensions for the QMP, it then follows that the tuples of squared, feasible TT-singular values form polyhedral cones. We further establish a connection to eigenvalue relations of sums of Hermitian matrices, which in turn are described by sets of interlinked, so-called honeycombs, as they have been introduced by Knutson and Tao. Besides a large class of universal, necessary inequalities as well as the vertex description for a special, simpler instance, we present a linear programming algorithm to check feasibility and a simple, heuristic algorithm to construct representations of tensors with prescribed, feasible TT-singular values in parallel. [ABSTRACT FROM AUTHOR]

Abstract We investigate the change in the multiplicities of the eigenvalues of a Hermitian matrix with a simple graph G , when edges are inserted into G or removed from G. We focus upon cases in which the multiplicity of the eigenvalue does not change due to inserting or removing edges incident to a vertex. Furthermore, we show how the change in the multiplicities of the eigenvalues occur, when two disjoint graphs are connected with one edge, based upon the status of the vertices that are connected. Lastly, we give the possible classifications of cut-edges in a graph and characterize the occurrence of each possible status. [ABSTRACT FROM AUTHOR]

Let A ∈ C n × n and A ˜ ∈ C n × n be two normal matrices with spectra { λ i } i = 1 n and { λ ˜ i } i = 1 n , respectively. 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 is no larger than the Frobenius norm of A ˜ − A . However, if either A or A ˜ is non-normal, this result does not hold in general. In this paper, we present several novel upper bounds for ( ∑ i = 1 n | λ ˜ π ( i ) − λ i | 2 ) 1 2 , provided that A is normal and A ˜ is arbitrary. Some of these estimates involving the “departure from normality” of A ˜ have generalized the Hoffman–Wielandt theorem. Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix. [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, we consider the eigenvalue variation for any perturbation of Hermitian matrices, and we obtain two perturbation bounds. The first bound always improves the existing bound, and the second bound also improves the existing one under a suitable condition. A simple example is given for comparing these bounds. [ABSTRACT FROM AUTHOR]

Let m , n ≥ 2 be positive integers and let H n be the set of n × n complex Hermitian matrices. We study linear maps ϕ : H m n → H m n satisfying det ⁡ ( A ⊗ B ) = det ⁡ ( ϕ ( A ⊗ B ) ) for all A ∈ H m , B ∈ H n . The connection of the problem to quantum information science is mentioned. [ABSTRACT FROM AUTHOR]

This work is concerned with approximating the smallest eigenvalue of a parameter-dependent Hermitian matrix A(μ) for many parameter values μ in a domain D ⊂ RP. The design of reliable and efficient algorithms for addressing this task is of importance in a variety of applications. Most notably, it plays a crucial role in estimating the error of reduced basis methods for parametrized partial differential equations. The current state-of-the-art approach, the so-called successive constraint method (SCM), addresses affine linear parameter dependencies by combining sampled Rayleigh quotients with linear programming techniques. In this work, we propose a subspace approach that additionally incorporates the sampled eigenvectors of A(μ) and implicitly exploits their smoothness properties. Like SCM, our approach results in rigorous lower and upper bounds for the smallest eigenvalues on D. Theoretical and experimental evidence is given to demonstrate that our approach represents a significant improvement over SCM in the sense that the bounds are often much tighter, at negligible additional cost. [ABSTRACT FROM AUTHOR]

For a Hermitian matrix A ∈ C p × p , given vectors x , y ∈ C p and for suitable functions f , the bilinear form y ⁎ f ( A ) x is estimated by extending the extrapolation method proposed by C. Brezinski in 1999. Families of one term and two term estimates e f , ν , ν ∈ C and e ˆ f , n , k , n , k ∈ Z , respectively, are derived by extrapolation of the moments of the matrix A . For the positive definite case, bounds for the optimal value of ν , which leads to an efficient one term estimate in only one matrix vector product, are derived. For f ( A ) = A − 1 , a formula approximating this optimal value of ν is specified. Numerical results for several matrix functions and comparisons are provided to demonstrate the effectiveness of the extrapolation method. [ABSTRACT FROM AUTHOR]

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]

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n × n matrix is a sequence ℓ 1 ℓ 2 ⋯ ℓ n where ℓ k is A , S , or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the k th entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero). Various techniques including the Schur complement are introduced to establish that certain subsequences such as NAN are forbidden in epr-sequences over fields of characteristic not two. Using probabilistic methods over fields of characteristic zero, it is shown that any sequence of A s and S s ending in A is attainable, and any sequence of A s and S s followed by one or more N s is attainable; additional families of attainable epr-sequences are constructed explicitly by other methods. For real symmetric matrices of orders 2, 3, 4, and 5, all attainable epr-sequences are listed with justifications. [ABSTRACT FROM AUTHOR]

Probability measures on the space of Hermitian matrices which are ergodic for the conjugation action of an infinite-dimensional unitary group are considered. It is established that the eigenvalues of random matrices distributed with respect to these measures satisfy the law of large numbers. The relationship between such models of random matrices and objects in free probability, freely infinitely divisible measures, is also established. [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]

Without 'positive definiteness' demanded in the present papers, the forward and reverse inequalities for Hadamard products of any number of invertible Hermitian matrices are obtained, and the sufficient and necessary conditions for the equations in these inequalities are given. As Hermitian positive matrices naturally satisfy the added constraints, these results generalize and improve the corresponding results in the present papers. Beyond that, with no demand of 'positive definiteness', these forward and backward inequalities are not determined mutually any longer. [ABSTRACT FROM AUTHOR]

We consider the problem of determining those undirected n -vertex graphs with a corresponding Hermitian matrix that admits only two distinct eigenvalues, with multiplicities k and n--k. After giving some general algebraic characterizations of these dual multiplicity graphs, we then prove two major graph theoretic necessary conditions on such graphs. Construction techniques are then developed, and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with some generalizations of the Hermitian and skew-Hermitian splitting (HSS) iteration for solving continuous Sylvester equations. The main contents we will introduce are the normal and skew-Hermitian splitting (NSS) iteration methods for the continuous Sylvester equations. It is shown that the new schemes can outperform the standard HSS method in some situations. Theoretical analysis shows that the NSS methods converge unconditionally to the exact solution of the continuous Sylvester equations. Moreover, we derive the upper bound of the contraction factor of the NSS iterations. Numerical experiments further show the effectiveness of our new methods. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the minimal rank of when X is Hermitian by applying singular value decomposition and some rank equalities of matrices, and obtain a representation of the minimal rank. Based on the representation, we obtain necessary and sufficient conditions for the matrix equation to have Hermitian solutions. [Copyright &y& Elsevier]

Abstract: Let be a positive semidefinite block matrix with square matrices M and N of the same order and denote . The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, If, in addition, K is Hermitian, then for any real number , while if K is skew-Hermitian, then for any real number , where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz. [Copyright &y& Elsevier]

We prove that a continuous map $\phi $ defined on the set of all $n\times n$ Hermitian matrices preserving order in both directions is up to a translation a congruence transformation or a congruence transformation composed with the transposition. [ABSTRACT FROM PUBLISHER]

Abstract: We investigate constant rank subspaces of symmetric and hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and hermitian matrices whose non-zero elements all have odd rank. [Copyright &y& Elsevier]

We investigate necessary and sufficient conditions for semiconvergence of a splitting for solving singular linear systems, where the coefficient matrix A is a singular EP matrix. When A is a singular Hermitian matrix, necessary and sufficient conditions for semiconvergence of P-regular splittings are given, which generalize known results. As applications, the necessary and sufficient conditions for semiconvergence of block AOR and SSOR iterative methods are derived. A numerical example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the additive, multiplicative and general splitting iteration methods for solving singular linear systems. When the coefficient matrix is Hermitian, the semiconvergence conditions are proposed, which generalize some results of Bai [Z.-Z. Bai, On the convergence of additive and multiplicative splitting iterations for systems of linear equations, J. Comput. Appl. Math. 154 (2003) 195–214] for the nonsingular linear systems to the singular systems. [Copyright &y& Elsevier]

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]

We further generalize the technique for constructing the Hermitian/skew-Hermitian splitting (HSS) iteration method for solving large sparse non-Hermitian positive definite system of linear equations to the normal/skew-Hermitian (NS) splitting obtaining a class of normal/skew-Hermitian splitting (NSS) iteration methods. Theoretical analyses show that the NSS method converges unconditionally to the exact solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the NSS iteration which is dependent solely on the spectrum of the normal splitting matrix, and is independent of the eigenvectors of the matrices involved. We present a successive-overrelaxation (SOR) acceleration scheme for the NSS iteration, which specifically results in an acceleration scheme for the HSS iteration. Convergence conditions for this SOR scheme are derived under the assumption that the eigenvalues of the corresponding block Jacobi iteration matrix lie in certain regions in the complex plane. A numerical example is used to show that the SOR technique can significantly accelerate the convergence rate of the NSS or the HSS iteration method. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F-norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above-mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new QR-like algorithm, symmetric squared QR (SSQR) method, that can be readily parallelized using commonly available parallel computational primitives such as matrix–matrix multiplication and QR decomposition. The algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts. We provide a rigorous convergence analysis of SSQR and derive structures for several of the important quantities generated by the algorithm. We also discuss various practical implementation issues such as stopping criteria and deflation techniques. We demonstrate the convergence behavior of SSQR using several numerical examples. [Copyright &y& Elsevier]

In this paper we propose the use of the eigensystem of complex adjacency matrices to analyze the structure of asymmetric directed weighted communication. The use of complex Hermitian adjacency matrices allows to store more data relevant to asymmetric communication, and extends the interpretation of the resulting eigensystem beyond the principal eigenpair. This is based on the fact, that the adjacency matrix is transformed into a linear self-adjoint operator in Hilbert space. Subgroups of members, or nodes of a communication network can be characterised by the eigensubspaces of the complex Hermitian adjacency matrix. Their relative ‘traffic-level’ is represented by the eigenvalue of the subspace, and their members are represented by the eigenvector components. Since eigenvectors belonging to distinct eigenvalues are orthogonal the subgroups can be viewed as independent with respect to the communication behavior of the relevant members of each subgroup. As an example for this kind of analysis the EIES data set is used. The substructures and communication patterns within this data set are described. [ABSTRACT FROM AUTHOR]

Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices A and B is related to the corresponding order between the squares A2 and B2, in the sense of the star partial ordering, the minus partial ordering, and the Lo¨wner partial ordering. In the present paper, possibilities of generalizing and strengthening their results are studied from two points of view: by widening the class of matrices considered and by replacing the squares by arbitrary powers. [Copyright &y& Elsevier]

An important theorem about the existence of principal submatrices of a Hermitian matrix whose graph is a tree, in which the multiplicity of an eigenvalue increases, was largely developed in separate papers by Parter and Wiener. Here, the prior work is fully stated, then generalized with a self-contained proof. The more complete result is then used to better understand the eigenvalue possibilities of reducible principal submatrices of Hermitian tridiagonal matrices. Sets of vertices, for which the multiplicity increases, are also studied. [ABSTRACT FROM AUTHOR]

We describe the set of all (α,n), for which the scalar complex matrix αIn is a sum of k idempotent Hermitian matrices, and get the minimal number of summands for each αIn. [Copyright &y& Elsevier]

For Hermitian matrices, whose graph is a given tree, the relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail. In the process, certain aspects of special vertices, whose removal results in an increase in multiplicity are investigated. [Copyright &y& Elsevier]

In this note we introduce a partial ordering in the set of complex Hermitian matrices which coincides with the well-known Löwner ordering when the considered matrices have the same number of negative eigenvalues. Some properties of the new ordering are investigated, and known results for shorted matrices are generalized. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices and modify shortcomings of some known inequalities for generalized eigenvalues in the related literature. [ABSTRACT FROM AUTHOR]