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Eigenvalue decomposition of quaternion Hermitian matrices is a crucial mathematical tool for color image reconstruction and recognition. Quaternion Jacobi method is one of the classical methods to compute the eigenvalues of a quaternion Hermitian matrix. Using quaternion Jacobi rotations, this paper brings forward an innovative method for the eigenvalue decomposition of dual quaternion Hermitian matrices. The effectiveness of the proposed method is confirmed through numerical experiments. Furthermore, a dual complex matrix representation for the color image is developed, and the dual quaternion Jacobi method is applied to the eigenvalue problems of dual complex Hermitian matrices. This approach achieves successful results in the color images reconstruction and recognition. Compared to the quaternion matrix representation of the color image, this approach makes computations more convenient when dealing with problems related to color image processing. [ABSTRACT FROM AUTHOR]

We prove new singular value inequalities involving submultiplicative and subadditive functions of matrices. Singular value inequalities for sums and direct sums of matrices are also given. [ABSTRACT FROM AUTHOR]

We introduce a new class of linear codes over a finite field associated to determinantal varieties in the space of hermitian matrices and determine their length, dimension and minimum distance along with the weight spectrum. [ABSTRACT FROM AUTHOR]

In this paper, the octonion matrix equation A X B = C is studied based on semi-tensor product of matrices. Firstly, we propose the left real element representation and the right real element representation of octonion. Then we obtain the expression of the least squares Hermitian solution to the octonion matrix equation A X B = C by combining these representations with H -representation of the special matrices. In addition, we also put forward the equivalent condition of existence and general expression of the Hermitian solution to the octonion matrix equation A X B = C. Finally, the validity and stability of our method is verified by numerical experiments. [ABSTRACT FROM AUTHOR]

A square matrix A over the field of complex numbers is said to be Hermitian if A = A ∗ , the conjugate transpose of A, while Hermitian matrices are known to be an important class of matrices. In addition to the definition, a Hermitian matrix can be characterized by some other matrix equalities. This fact can be described in the implication form f (A , A ∗) = 0 ⇔ A = A ∗ , where f (·) denotes certain ordinary algebraic operation of A and A ∗ . In this note, we show two special cases of the equivalent facts: A A ∗ A = A ∗ A A ∗ ⇔ A 3 = A A ∗ A ⇔ A = A ∗ without assuming the invertibility of A through the skillful use of decompositions and determinants of matrices. Several consequences and extensions are presented to a selection of matrix equalities composed of multiple products of A and A ∗ . [ABSTRACT FROM AUTHOR]

Matrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no polynomial-time approximation scheme (unless P = NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmically few qubits. [ABSTRACT FROM AUTHOR]

This paper investigates the interplay between the algebraic properties of the rings, the combinatorial properties of their corresponding zero-divisor graphs, and the associated Hermitian matrix of such graphs. For a finite ring R, its zero-divisor graph may contain both directed edges and undirected edges; such graphs are called mixed graphs. The Hermitian matrices of mixed graphs are natural generalizations of the adjacency matrices of undirected graphs. In this paper, we completely determine the structure and the Hermitian eigenvalues of the zero-divisor graph Γ (D × R) by using the structure and the Hermitian eigenvalues of the zero-divisor graph Γ (R) . As applications, we investigate Γ (D × R) for some special R and extend some known results on this topic. [ABSTRACT FROM AUTHOR]

A basic theorem in linear algebra says that if the eigenvalues and the diagonal entries of a Hermitian matrix A are ordered as λ 1 ≤ λ 2 ≤... ≤ λ n and a 1 ≤ a 2 ≤... ≤ a n , respectively, then λ 1 ≤ a 1 . We show that for some special classes of Hermitian matrices this can be extended to inequalities of the form λ k ≤ a 2 k - 1 , k = 1 , 2 ,... , ⌈ n 2 ⌉ . [ABSTRACT FROM AUTHOR]

Sylvester-like matrix equations are encountered in many areas of control engineering and applied mathematics. In this paper, we construct some necessary and sufficient conditions for the system of Hermitian mixed type generalized Sylvester matrix equations to have a solution. The closed form formula to compute the general solution is also established when solvability conditions are satisfied. An algorithm and a numerical example are provided to validate our findings. [ABSTRACT FROM AUTHOR]

In this paper, we apply the semi-tensor product of matrices and the real vector representation of quaternion matrix to solve quaternion matrix equations for the first time. We derive the expressions of the least squares Hermitian, Persymmetric and Bisymmetric solution for the quaternion matrix equation A X B = C , respectively. Furthermore, we also put forward the equivalent conditions of existence and general expression of the Hermitian, Persymmetric and Bisymmetric solution to the quaternion matrix equation A X B = C , respectively. Finally, the validity of our method is verified by numerical experiments with different dimensions. [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]

Inspired by the work of Zhou (Des Codes Cryptogr 88:841–850, 2020) based on the paper of Schmidt (J Algebraic Combin 42(2):635–670, 2015), we investigate the equivalence issue of maximum d-codes of Hermitian matrices. More precisely, in the space H n (q 2) of Hermitian matrices over F q 2 we have two possible equivalences: the classical one coming from the maps that preserve the rank in F q 2 n × n , and the one that comes from restricting to those maps preserving both the rank and the space H n (q 2) . We prove that when d < n and the codes considered are maximum additive d-codes and (n - d) -designs, these two equivalence relations coincide. As a consequence, we get that the idealisers of such codes are not distinguishers, unlike what usually happens for rank metric codes. Finally, we deal with the combinatorial properties of known maximum Hermitian codes and, by means of this investigation, we present a new family of maximum Hermitian 2-code, extending the construction presented by Longobardi et al. (Discrete Math 343(7):111871, 2020). [ABSTRACT FROM AUTHOR]

A thick-restart Lanczos type algorithm is proposed for Hermitian J-symmetric matrices. Since Hermitian J-symmetric matrices possess doubly degenerate spectra or doubly multiple eigenvalues with a simple relation between the degenerate eigenvectors, we can improve the convergence of the Lanczos algorithm by restricting the search space of the Krylov subspace to that spanned by one of each pair of the degenerate eigenvector pairs. We show that the Lanczos iteration is compatible with the J-symmetry, so that the subspace can be split into two subspaces that are orthogonal to each other. The proposed algorithm searches for eigenvectors in one of the two subspaces without the multiplicity. The other eigenvectors paired to them can be easily reconstructed with the simple relation from the J-symmetry. We test our algorithm on randomly generated small dense matrices and a sparse large matrix originating from a quantum field theory. [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]

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]

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]

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 shall consider a double infinite, Hermitian, complex entry matrix $${A=[a_{x,y}]_{x,y\in\mathbb{Z}}}$$ . In the present note we give a criterion, expressed in terms of the entries of the matrix, for the corresponding symmetric operator defined on compactly supported sequences, to be essentially self-adjoint in the space $${\ell_2(\mathbb{Z})}$$ . Roughly speaking, assuming that x denotes the row number, we require that: (1) there exist $${\gamma\in[0,1)}$$ and n > 0 for which the entries that are at distance larger than $${n(|x|^2+1)^{\gamma/2}}$$ from the diagonal vanish and (2) the $${\ell^1}$$ norm of the xth row grows slower that $${|x|^{\gamma-1}}$$ , as $${|x|\to+\infty}$$ . [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]

Copyright of SCIENCE CHINA Information Sciences is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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]

This paper discusses the problem of two-dimensional direction of arrival estimation of coherent sources for acoustic vector sensors array. Compared with subspace-based methods, such as root multiple signal classification and estimation of signal parameters via rotational invariance technique, the propagator method (PM) has lower computational complexity. However, only in high-snapshots situation, can the PM algorithm enjoy a better estimation performance. Besides, all of these algorithms mentioned above cannot work for coherent sources. In this paper, we combine PM algorithm with Toeplitz Hermitian matrix reconstruction, and propose an improved algorithm, which works well in the case of coherent signals and a single snapshot. Furthermore, the proposed method can achieve automatically paired two-dimensional angle estimation. Simulation results verify that the proposed method has the better angle performance and less computational complexity than spatial smoothing methods. [ABSTRACT FROM AUTHOR]

Full-rank block LDL decomposition of a Hermitian n× n block matrix A is examined, where the iterative procedure evaluating the sub-matrices appearing in L and D is provided. This factorization is used to evaluate the inverse and Moore-Penrose inverse of a Hermitian n× n block matrix. The method for the calculation of the Moore-Penrose inverse of an arbitrary 2×2 block matrix is also provided. Therefore, matrix products A A and AA and the corresponding full-rank block LDL factorizations are observed. Also, a simple explicit formulae calculating the solution vector components of the normal system of equations is stated, where the LDL decomposition of the system matrix is done. [ABSTRACT FROM AUTHOR]

This paper is concerned with a modified version of the generalization of Hermitian and skew-Hermitian splitting iteration (MGHSS) to solve the non-Hermitian positive definite linear systems. The corresponding inexact MGHSS (IMGHSS) method is developed by employing some Krylov subspace methods as its inner process. Numerical examples are reported to confirm the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

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]

We introduce a simultaneous decomposition for a matrix triplet ( A, B, C), where A=± A and (⋅) denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A− BXC±( BXC) with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A− BXC−( BXC) with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D− CXC subject to Hermitian solutions of a consistent matrix equation AXA= B, as well as the extremal ranks and inertias of the Hermitian Schur complement D− B A B with respect to a Hermitian generalized inverse A of A. Various consequences of these extremal ranks and inertias are also presented in the paper. [ABSTRACT FROM AUTHOR]

In this note, for a system of q matrix equations of the form we consider the problem of the existence of Hermitian nonnegative-definite solutions. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with q = 2 to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who proposed a revised version of Young et al. [2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement. [ABSTRACT FROM AUTHOR]

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]

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix. [ABSTRACT FROM AUTHOR]

Let D be any division ring with an involution, ℋ n ( D) be the space of all n × n hermitian matrices over D. Two hermitian matrices A and B are said to be adjacent if rank( A − B) = 1. It is proved that if ϕ is a bijective map from ℋ n ( D)( n ≥ 2) to itself such that ϕ preserves the adjacency, then ϕ −1 also preserves the adjacency. Moreover, if ℋ n ( D ≠ $${\fancyscript S}$$ 3( $${\fancyscript F}$$ 2), then ϕ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe–Xian is answered for geometry of symmetric and hermitian matrices. [ABSTRACT FROM AUTHOR]

Given a graph G with characteristic polynomial ϕ( t), we consider the ML-decomposition ϕ( t) = q 1( t) q 2( t)2 ... q m ( t)m, where each q i ( t) is an integral polynomial and the roots of ϕ( t) with multiplicity j are exactly the roots of q j ( t). We give an algorithm to construct the polynomials q i ( t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E( G) = $$\mathop \sum \limits_{i = 1}^n $$ |λi| of G, where λ1, λ2, ..., λn are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we establish a matrix inequality and its equivalent form. Applying the results, some matrix inequalities involving KhatriRao products of positive semi-definite matrices are generalized. [ABSTRACT FROM AUTHOR]

Focuses on the stability of linear stochastic finite-difference systems with almost-periodic coefficients. Reduction of the stability problem of solutions of deterministic linear finite-difference equation; Establishment of a condition for exponential stability; Validity of the criterion of asymptotic mean-square stability.

Denote the set of n×n complex Hermitian matrices by H n . A pair of n×n Hermitian matrices ( A, B) is said to be rank-additive if rank ( A+B)=rank A+rank B. We characterize the linear maps from H n into itself that preserve the set of rank-additive pairs. As applications, the linear preservers of adjoint matrix on H n and the Jordan homomorphisms of H n are also given. The analogous problems on the skew Hermitian matrix space are considered. [ABSTRACT FROM AUTHOR]

Let D be a division ring which possesses an involution a → α . Assume that $$F = \{ a \in D|a = \overline a \} $$ is a proper subfield of D and is contained in the center of D. It is pointed out that if D is of characteristic not two, D is either a separable quadratic extension of F or a division ring of generalized quaternions over F and that if D is of characteristic two, D is a separable quadratic extension of F. Thus the trace map Tr: D → F, a → a + a is always surjective, which is formerly posed as an assumption in the fundamental theorem of n×n hermitian matrices over D when n ≥ 3 and now can be deleted. When D is a field, the fundamental theorem of 2 × 2 hermitian matrices over D has already been proved. This paper proves the fundamental theorem of 2×2 hermitian matrices over any division ring of generalized quaternions of characteristic not two [ABSTRACT FROM AUTHOR]

Solvability conditions are examined for the matrix equation $$X^{ - 1} X* = A$$ , which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated. [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order $$r$$ for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian $$((n + r) \times (n + r))$$ -matrix with a given $$(n \times n)$$ -block $$A_{11} $$ in block $$(2 \times 2)$$ -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian $$(n \times n)$$ -matrix $$B$$ of rank not greater than $$r$$ so that the obtained matrix, being added to a given Hermitian $$(n \times n)$$ -matrix $$A$$ , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems. [ABSTRACT FROM AUTHOR]

We investigate congruence classes of m-tuples of points in the quaternionic elliptic space ℍ P n. We establish a canonical bijection between the set of congruence classes of m-tuples of points in ℍ P n and the set of equivalence classes of positive semidefinite Hermitian m× m matrices of rank at most n+1 with the 1's on the diagonal. We show that with each m-tuple of points in ℍ P n is associated a tuple of points on the real unit sphere S2. Then we get that the congruence class of an m-tuple of points in ℍ P n is determined by the congruence classes of all its triangles and by the direct congruence class of the associated tuple on the sphere S2 provided that no pair of points of the m-tuple has distance π/2. Finally we carry out the same kind of investigation for the quaternionic hyperbolic space ℍ H n. Most of the results are completely analogous, although there are also some interesting differences. [ABSTRACT FROM AUTHOR]

We give a simple proof of a determinantal inequality due to Marcus (1976). [ABSTRACT FROM AUTHOR]

Copyright of Trabajos de Estadística is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article discusses the measurement of the orthogonal matrix polynomials with periodic coefficients of recurrence relations. It uses the diagonalization of the matrix transform to determine the matrices. It presents mathematical theorems that explain the problem of reconstructing the measure of a periodic sequence of coefficients.

The article discusses the proof of Thompson's determinantal inequality and tackles the conjugate transpose of a matrix or a vector X.

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]