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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]

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]

In this paper, two new algorithms are proposed for non-orthogonal joint matrix diagonalization under Hermitian congruence. The idea of these two algorithms is based on the so-called Jacobi algorithm for solving the eigenvalues problem of Hermitian matrix. The algorithms are then called ‘general Jabobi-like diagonalization’ algorithms (GERALD). They are based on the search of two complex parameters by the minimization of a quadratic criterion corresponding to a measure of diagonality. Lastly, numerical simulations are conducted to illustrate the effective performances of the GERALD algorithms. [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 classical problem of characterizing the graphs whose eigenvalues lie in a given interval may date back to the work of J.H. Smith in 1970. Especially, the research on graphs with smallest eigenvalues not less than −2 has attracted widespread attention. Mixed graphs are natural generalizations of undirected graphs. In this paper, we completely characterize the mixed graphs with smallest Hermitian eigenvalue greater than − 5 + 1 2. In fact, we found three infinite classes of mixed graphs and 30 scattered mixed graphs. [ABSTRACT FROM AUTHOR]

Fundamental theorem of geometry of hermitian matrices characterizes all bijective maps on the space of all hermitian matrices, which preserve adjacency in both directions. In this and subsequent paper we characterize maps on the set of all invertible hermitian matrices over a finite field, which preserve adjacency in one direction. In this first paper it is shown that maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are necessarily bijective, so the corresponding graph on invertible hermitian matrices, where edges are defined by the adjacency relation, is a core. Besides matrix theory, the proof relies on results from several other mathematical areas, including spectral and chromatic graph theory, and finite geometry. [ABSTRACT FROM AUTHOR]

Maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are characterized. It is shown that such maps form a group that is generated by the maps A ↦ P A P ⁎ , A ↦ A σ , and A ↦ A − 1 , where P is an invertible matrix, P ⁎ is its conjugate transpose, and σ is an automorphism of the underlying field. Bijectivity of maps is not an assumption but a conclusion. Moreover, adjacency is assumed to be preserved in one direction only. [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]

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]

The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal { k + 1 } -potent matrix are considered. First, we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them. Then, when such a solution exists, an expression for the solution to the corresponding optimal approximation problem is obtained. [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]

We employ the implicit finite difference scheme with the shifted Grünwald formula to discretize the fractional diffusion equations with constant coefficients. The coefficient matrix possesses the positive definite Toeplitz-like structure, so we can use the Hermitian and skew-Hermitian splitting method for solving the system. Krylov subspace methods with circulant preconditioners such as Strang’s and T. Chan’s preconditioners are proposed to solve each subsystem via using the fast Fourier transforms (FFTs). Moreover, we present convergence analysis and prove the spectrum of the preconditioned matrices to be clustered around 1. Superlinear convergence rates of the proposed algorithms are obtained. Numerical results illustrate the effectiveness and robustness of circulant preconditioners. [ABSTRACT FROM AUTHOR]

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]

For a positive integer $n\geq 2$, let $M_{n}$ be the set of $n\times n$ complex matrices and $H_{n}$ the set of Hermitian matrices in $M_{n}$. We characterize injective linear maps ${\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}$ satisfying $$\begin{eqnarray}\text{rank}(A_{1}\otimes \cdots \otimes A_{l})=1\Longrightarrow \text{rank}({\it\phi}(A_{1}\otimes \cdots \otimes A_{l}))=1\end{eqnarray}$$ for all $A_{k}\in H_{m_{k}}$, $k=1,\dots ,l$, where $l,m_{1},\dots ,m_{l}\geq 2$ are positive integers. The necessity of the injectivity assumption is shown. Moreover, the connection of the problem to quantum information science is mentioned. [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]

This paper is concerned with several variants of the Hermitian and skew-Hermitian splitting iteration method to solve a class of complex symmetric linear systems. Theoretical analysis shows that several Hermitian and skew-Hermitian splitting based iteration methods are unconditionally convergent. Numerical experiments from an n-degree-of-freedom linear system are reported to illustrate the efficiency of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We prove a generalization of Samuelson's inequality for higher order central moments. Bounds for the eigenvalues are obtained when a given complex n × n matrix has real eigenvalues. Likewise, we discuss bounds for the roots of polynomial equations. [ABSTRACT FROM AUTHOR]

We obtain a new matrix inequality for an arbitrary density matrix of composite/noncomposite qudit systems including a single-qudit state. For bipartite systems, this inequality coincides with a known entropic inequality like the subadditivity condition. The examples of two-qubit system and qudit with j = 3 /2 are discussed. [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 obtain a new criterion for the complete indeterminacy of the classical matrix problem and of the Nevanlinna-Pick matrix problem in terms of the convergence of two matrix series. The elements of these series are positive matrices and are expressed by analogs of the classical Schur parameters. [ABSTRACT FROM AUTHOR]

We unify and generalize several inequalities for the number wk of walks of length k in graphs, and for the entry sum of matrix powers. First, we present a weighted sandwich theorem for Hermitian matrices which generalizes a matrix theorem by Marcus and Newman and which further generalizes our former unification of inequalities for the number of walks in undirected graphs by Lagarias et al. and by Dress and Gutman. The new inequality uses an arbitrary nonnegative weighting of the indices (vertices) which allows to apply the theorem to index (vertex) subsets (i.e., inequalities considering the number wk(S,S) of walks of length k that start at a vertex of a given vertex subset S and that end within the same subset). We also deduce a stronger variation of the sandwich theorem for the case of positive-semidefinite Hermitian matrices which generalizes another inequality of Marcus and Newman. Further, we show a similar theorem for nonnegative symmetric matrices which is another unification and generalization of inequalities for walk numbers by Erdős and Simonovits, by Dress and Gutman, and by Ilić and Stevanović. In the last part, we generalize lower bounds for the spectral radius of adjacency matrices and upper bounds for the energy of graphs. [ABSTRACT FROM AUTHOR]

Given an n × n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0 , 1 , … , n , a 1 in the k th position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such attainable sequences determined for all n over any field with characteristic 2. A complete list of attainable sequences for real symmetric matrices of order 7 is reported. [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]

Using the monotonicity of relative entropy of composite quantum systems, we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Examples of qutrit state inequalities and the 'qubit portrait' bound for the distance between the qutrit states are considered in explicit form. [ABSTRACT FROM AUTHOR]

In this paper, we present several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing interior-point methods (IPMs) for semidefinite optimization (SDO). [ABSTRACT FROM AUTHOR]

Recently, Gnutzmann and Smilansky [5] presented a formula for the bond scattering matrix of a graph with respect to an Hermitian matrix. We present another proof for this formula by a technique use in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilansky's formula to a regular covering of a graph. Finally, we define an L -function of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilansky's formula to a regular covering of a graph by using its L -functions. [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]

This paper investigates the validity of the random-volume-over-ground (RVoG) scattering model assumption for forest scattering on polarimetric interferometric synthetic aperture radar (PolInSAR) data. The model makes some assumptions about the data and the structure of coherency matrices, namely, the equality of the polarimetric covariance matrices and the affine equivalence of the contracted polarimetric interferometric covariance matrix with a Hermitian matrix. The proposed methodology is divided into two main steps. First, invertible affine transforms (ATs) are studied and proposed as a tool to operate with coherence regions. Based on this analysis, the concept of the trace matrix is introduced as its rank depends on the RVoG model assumption validity. Then, with the objective to consider the effects of speckle noise, we consider a maximum-likelihood (ML) framework, on the hypothesis of data distributed according to the complex Gaussian distribution. Hence, we define the ML estimator (MLE) of the PolInSAR coherency matrix according to the RVoG model assumption and the generalized likelihood ratio test of the model. The validity tests and the MLE are analyzed in terms of simulated and real PolInSAR data, considering P-band and L-band data over tropical and boreal forests. [ABSTRACT FROM PUBLISHER]

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]

SUMMARY In this paper, we present some new interlacing properties about the bounds of the eigenvalues for rank-one modification of Hermitian matrix, whose eigenvalues are different and spectral decomposition also needs to be known. Numerical examples demonstrate the efficiency of the proposed method and support our theoretical results.Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices. We also study the related question: given two matrices, which matrix under the Cayley transform yields the Kronecker product of their Cayley transforms. [Copyright &y& Elsevier]

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]

In this paper, the n-th root of a matrix is defined, and the explicit form of n-th root of an Hermitian matrix is given. A new method for diagonalizing quadratic Hamiltonians is proposed. Also, a class of quantum operators is induced by the linear transformation in configuration space, and its unitary properties and transformation behavior are studied. Our new method based on n-th root of matrices can develop the mathematical methods of quantum mechanics and quantum optics, and can also be applied to engineering, quantum optics and quantum fields states with squeezing properties, as well as the binomial field states. [ABSTRACT FROM AUTHOR]

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]

If a convex body C in …n has modular and irreducible face lattice and C is not strictly convex, there is a face-preserving homeomorphism from C to a set of positive-semidefinite Hermitian matrices of trace 1 over …, … or …, or C has dimension 8, 14 or 26. [ABSTRACT FROM AUTHOR]

Abstract: We show some majorization inequalities and apply them to derive norm, eigenvalue, singular value, and trace inequalities of matrices. We also present a generalization and a different proof of a recent result on majorization concerning partitioned Hermitian positive semidefinite matrices. [Copyright &y& Elsevier]

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]

Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely many solutions were obtained in Hamburger's papers on the classical moment problem. In this paper, we obtain an analog of Hamburger's criterion for the matrix moment problem. [ABSTRACT FROM AUTHOR]

Abstract: We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices. [Copyright &y& Elsevier]

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 derive some properties of complex Hessenberg matrices and use the relevant normal matrix cases to examine the lengths of Ritz vectors in the rounding error analysis of the Lanczos tridiagonalization process. This question is important for the computational use of the process and has already been studied for the real symmetric matrix case, but because of its intricate and unedifying nature, part of the theory was never submitted to scientific journals. We develop a new and more palatable theory which also applies to Lanczos processes adapted to any form of normal matrix with collinear eigenvalues such as a Hermitian or skew-Hermitian matrix. The nonnormal matrix properties are intended to help in the analysis of the unsymmetric Lanczos process. [ABSTRACT FROM AUTHOR]

Abstract: Estimating upper bounds of the spectrum of large Hermitian matrices has long been a problem with both theoretical and practical significance. Algorithms that can compute tight upper bounds with minimum computational cost will have applications in a variety of areas. We present a practical algorithm that exploits k-step Lanczos iteration with a safeguard step. The k is generally very small, say 5–8, regardless of the large dimension of the matrices. This makes the Lanczos iteration economical. The safeguard step can be realized with marginal cost by utilizing the theoretical bounds developed in this paper. The bounds establish the theoretical validity of a previous bound estimator that has been successfully used in various applications. Moreover, we improve the bound estimator which can now provide tighter upper bounds with negligible additional cost. [Copyright &y& Elsevier]