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In 2017, M. Lin formulated two conjectures concerning determinantal inequalities for positive semi-definite matrices A and B , and which can be stated as follows det ⁡ (A 2 + | A B | p) ≥ det ⁡ (A 2 + | B A | p) for p ≥ 0 and det ⁡ (A 2 + | A B | p) ≥ det ⁡ (A 2 + A p B p) for 0 ≤ p ≤ 2. The main goal of this paper is to confirm the first conjecture in a slightly more general setting namely in the case when A and B are Hermitian, and also to prove the second conjecture when 0 ≤ p ≤ 4 3. Various related inequalities are then presented and we conclude with an open log-majorization question. [ABSTRACT FROM AUTHOR]

For n × n complex matrices, we show many inequalities under the unitarily invariant norm. [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 singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E (A) is the sum of its singular values. The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E (A) ≥ 2 E (D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E (H) is an edge cut of G , then E (H) ≤ E (G) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E (G) ≥ 4 m + n (n − 2) | det ⁡ (A) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E (G) given in Oboudi (2019) [14]. [ABSTRACT FROM AUTHOR]

Matrices in R n × n with determinant equal to ∑ i ≤ n μ (i) , where μ represents the Möbius function, have been studied for decades. The motivation to study such matrices is the close connection that they have to the prime number theorem, Dirichlet convolution, and related concepts. We introduce two parameterized families of bordered Hermitian matrices that possess similar properties. Each family is comprised of matrices M s ∈ C (n − 1) × (n − 1) that satisfy det ⁡ M 0 = ∑ i ≤ n | μ (i) | , det ⁡ M 1 = (∑ i ≤ n μ (i)) 2 , and we show det ⁡ M s is a quadratic polynomial in s. We apply the Cauchy interlacing theorem to show that, for each matrix in one of the families, the product of all of the subdominant eigenvalues is bounded above by 6 / π 2 + O (n − 1 / 2). [ABSTRACT FROM AUTHOR]

Let D denote the smallest disc containing all eigenvalues of the matrix A. Without knowing the eigenvalues of A, we can estimate the spread of A and the radius of D. Some new bounds for the radius of D and the spread of A are given. These bounds involve the entries of A. Also sufficient conditions for equality are obtained for some inequalities. New proofs of some known results are presented, too. [ABSTRACT FROM AUTHOR]

Let S be an n-by-n, nonsingular, and Hermitian matrix. A square complex matrix Q is said to be S-unitary if Q*SQ = S. An S-unitary matrix Q is said to be elementary if rank(Q -- I) = 1. It is known what form every elementary S-unitary can take, and that every S-unitary can be written as a product of elementary S-unitaries. In this paper, we determine the Jordan canonical form of a product of two elementary S-unitaries. [ABSTRACT FROM AUTHOR]

We present verification methods to compute error bounds for all eigenvectors of a Hermitian matrix as well as for all singular vectors of a rectangular real or complex matrix. In case of clusters these are bounds for an orthonormal basis of the invariant subspace or singular vector space, respectively. Individual error bounds for all eigenvalues and singular values including clustered and/or multiple ones are computed as well. The computed bounds do contain the true result with mathematical certainty, and the algorithms apply to interval data as well. In that case the computed bounds are true for every real/complex matrix within the tolerances. The computational complexity to compute inclusions of all eigen/singular pairs of an n × n matrix or m × n matrix is O (n 3) or O (m n 2) operations, respectively. [ABSTRACT FROM AUTHOR]

The paper develops techniques to present an explicit diagonalization of 3 × 3 -Hermitian matrices over K -octonions O (K) via the K -exceptional group F 4 (K) provided K is a real closed field. Then, an action of F 4 (K) on the K -Cayley plane O (K) P 2 is studied to show a bijection F 4 (K) / Spin (9 , K) ⟶ ≈ O (K) P 2 for the stabilizer Spin (9 , K) of the matrix E 11 ∈ O (K) P 2 provided K is a formally real Pythagorean field. [ABSTRACT FROM AUTHOR]

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product. [ABSTRACT FROM AUTHOR]

We establish the converse of Weyl's eigenvalue inequality for additive Hermitian perturbations of a Hermitian matrix. [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]

A finite computational process using only arithmetical operations is called a rational algorithm. Presently, there is no known rational algorithm for checking congruence between arbitrary complex matrices A and B. The situation may be different if A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. In this publication, we propose a rational algorithm for checking congruence between matrices A and B that are square roots of Hermitian matrices. [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]

Answering a question of Erdös, Komlós proved in 1968 that almost all n×n Bernoulli matrices are nonsingular as n→∞. In this paper, we offer a new perspective on the question of Erdös by studying n×n matrices with prime number entries in an almost all sense. Precisely, it is shown that, as x→∞, the probability of randomly choosing a nonsingular n×n matrix among all n×n matrices with prime number entries that are ≤x is 1. If A is a unitary matrix, then it is well known that |detA|=1. However, the converse is far from being true. As a remedy of this defect, we search for necessary and sufficient conditions for being a unitary matrix by teaming up determinant with trace. In this way, we are led to simple characterizations of unitary matrices in the set of normal matrices. The question of which nonsingular commuting complex matrices with real eigenvalues have the same characteristic polynomial is formulated via determinant and trace conditions. Finally, through a study of eigenvectors, we obtain new characterizations of Hermitian and normal matrices. Our approach to proving these results benefits from a modular interpretation of nonsingularity and the spectral theorem for normal operators together with equality cases of classical inequalities such as the arithmetic-geometric mean inequality and the Cauchy-Schwarz inequality. [ABSTRACT FROM AUTHOR]

The proof of the asymptotic quadratic convergence is provided for the parallel two-sided block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices. The discussion covers the case of well-separated eigenvalues as well as clusters of eigenvalues. Having p processors, each parallel iteration step consists of zeroing 2p off-diagonal blocks chosen by dynamic ordering with the aim to maximize the decrease of the off-diagonal Frobenius norm. Numerical experiments illustrate and confirm the developed theory. [ABSTRACT FROM AUTHOR]

In this paper, based on the LU decomposition, we propose three non-orthogonal Jacobi-like alternating iterative algorithms with two strategies for solving the joint diagonalization problem of a set of Hermitian matrices. In this kind of algorithm, each transformation includes one upper triangular iterative step and one lower triangular iterative step, and each step involves one parameter. The optimal parameter of each step is derived analytically. The convergence of our proposed algorithms is proven. According to this convergence analysis, the existing GNJD algorithm is revisited. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed algorithms in comparison with existing ones. [ABSTRACT FROM AUTHOR]

We prove that every continuous map acting on the four-dimensional Minkowski space and preserving light cones in one direction only is either a Poincaré similarity, that is, a product of a Lorentz transformation and a dilation, or it is of a very special degenerate form. [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]

The eigenspectrum μ1≥ μ2 ≥...≥ μm with the middle eigenvalues μh and μl , where h = [(m+1)/2] and l = d[(m+1)/2] of simple connected graph G' with m number of vertices contribute significantly in the H..uckel Molecular Theory. The HOMO-LUMO gap ▵G' is defined as ▵G' = μh -μl subject to the condition that the number of electrons are in one to one correspondence with the number of vertices. In this article, the upper bounds for the HOMO-LUMO gap corresponding to the connected graphs of nanotube TUC4C8(S) and C4C8 nanotorus by using matrix theory are estimated. [ABSTRACT FROM AUTHOR]

We prove some inequalities involving fourth central moment of a random variable that takes values in a given finite interval. Both discrete and continuous cases are considered. Bounds for the spread are obtained when a given $$n\times n$$ complex matrix has real eigenvalues. Likewise, we discuss bounds for the spans of polynomial equations. [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 1893 Jacques Hadamard introduced the famous inequality concerning the determinant of the Gram matrix. The generalizations of this inequality are the main subject of considerations presented in this paper. Calculations executed for the gramian of order three revealed that the description of gramian hides a great potential enabling to strengthen essentially the Hadamard inequality. The research, undertaken in this direction, resulted in discovering the more subtle versions of the Hadamard inequality, also by involving the “language” of the positive semi-definite Hermitian matrices. [ABSTRACT FROM AUTHOR]

We propose a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems. The proposed method uses complex moments to extract the eigencomponents of interest from a random matrix and uses the Rayleigh–Ritz procedure to project a given eigenvalue problem into a reduced eigenvalue problem. The complex moment is given by contour integral and approximated using numerical quadrature. We split the error in the complex moment into the truncation error of the quadrature and rounding errors and evaluate each. This idea for error evaluation inherits our previous Hankel matrix approach, whereas the proposed method enables verification of eigenvectors and requires half the number of quadrature points for the previous approach to reduce the truncation error to the same order. Moreover, the Rayleigh–Ritz procedure approach forms a transformation matrix that enables verification of the eigenvectors. Numerical experiments show that the proposed method is faster than previous methods while maintaining verification performance and works even for nearly singular matrix pencils and in the presence of multiple and nearly multiple eigenvalues. [ABSTRACT FROM AUTHOR]

Although Selberg-type single positive definite symmetric matrices gamma and beta integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe [2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However, several other types of Selberg-type integrals appear to have been neglected in the literature. Thus e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta densities, their noncentral counterparts, etc, have not been explored as yet. The present paper records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6]. [ABSTRACT FROM AUTHOR]

This paper proposes neural network for computing the eigenvectors of Hermitian matrices. For the eigenvalues of Hermitian matrices, we establish an explicit representation for the solution of the neural network based on Hermitian matrices and analyze its convergence property. We also consider to compute the eigenvectors of skew-symmetric matrices and skew-Hermitian matrices, corresponding to the imaginary maximal or imaginary minimal eigenvalue, based on the neural network for computing the eigenvectors of Hermitian matrices. Numerical examples are given to illustrate our theoretical result are valid. [ABSTRACT FROM AUTHOR]

A new iterative method for computing the polar decomposition of any rectangular complex matrix is presented and analyzed. The study of the convergence shows that this method has order of convergence six. Some numerical tests confirm the theoretical results and allow us to compare the proposed iterative scheme with other known ones. [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]

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]

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]

We find a method for symbolic estimation of a minimal (maximal) distance between (different) eigenvalues of a Hermitian operator (of a Hermitian matrix) using Hankel matrices formalism. [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]