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

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]

Summary: The hybrid precoding problem is considered a Frobenius norm reduction problem for the narrowband channel in a millimeter wave (mmWave) with multiple inputs and outputs (mmWave MIMO). This work proposes a hybrid distributed online and alternating convex optimization (HDO‐ACO) algorithm to improve hybrid precoding (HP), interpreted as a trace minimization problem. HDO‐ACO alternately determines the digital and analog precoders to reduce the trace by keeping the other constant. Initially, HDO‐ACO uses Lagrange's method to determine the digital precoding subproblem. Then, it uses the integrated distributed online convex optimization and alternating minimization algorithm in the analog precoder design. In the HP method, the digital and analog precoders are iteratively updated until the highest number of iterations or convergence is reached. But this hybrid precoding method requires an initial analog precoding matrix input to begin the iteration. The algorithms converge gradually and fall into a suboptimal solution when the initial analog precoding matrix is set randomly. Hence, an initial value acceleration‐based heuristic approach is used in the HDO‐ACO alternating minimization algorithm that calculates the initial feasible value of an alternating minimization method using channel conditions. The simulation results of the proposed HDO‐ACO algorithm are presented under bit error rate (BER), spectral efficiency (SE), and convergence behavior by comparing it with modified block coordinate descent–HP (MBCD‐HP), manifold optimization–alternating minimization (MO‐AltMin), and semi‐definite relaxation‐based alternating optimization (SDR‐AO). The proposed HDO‐ACO attains maximum SE and less BER than other hyper‐precoding designs. [ABSTRACT FROM AUTHOR]

The Cayley transform of a matrix A , C (A) = (I + A) − 1 (I − A) , has been investigated in the contexts of skew-symmetry, eigenvalue mapping and as a factorization technique for several matrix positivity classes. Here, we extend the discussion to Cayley transforms of almost skew-Hermitian matrices, positive definite matrices, semipositive matrices, as well as subclasses of the M -matrices, H -matrices and their inverses. [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]

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]

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.

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]

Batched matrix computing problems are widely existed in scientific computing and engineering applications.With rapid performance improvements,GPU has become an important tool to solve such problems.The eigenvalue decomposition belongs to the two-sided decomposition and must be solved by the iterative algorithm.Iterative numbers for different matrices can be varied.Therefore,designing eigenvalue decomposition algorithms for batched matrices on the GPU is more challenging than designing batched algorithms for the one-sided decomposition,such as LU decomposition.This paper proposes batched algorithms based on the Jacobi algorithms for eigenvalue decomposition of Hermitian matrices.For matrices that cannot reside in shared memory wholly,the block technique is used to improve the arithmetic intensity,thus improving the use of GPU resources.Algorithms presented in this paper run completely on the GPU,avoiding the communication between the CPU and GPU.Kernel fusion is adopted to decrease the overhead of launching kernel and global memory access.Experimental results on V100 GPU show that our algorithms are better than existing works.Performance evaluation results of the Roofline model indicate that our implementations are close to the upper bound,approaching 4.11TFLOPS.

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]

We examine to what extent a projection theorem is possible in the non-commutative (free) setting. We first review and extend some results that count against a full free projection theorem. We then obtain a weak version of the projection theorem: projections along linear and separated variables yield semialgebraically parametrised free semi-algebraic sets. [ABSTRACT FROM AUTHOR]

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

For any simple connected graph G of order n, having eigen spectrum μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n with middle eigenvalues μ H and μ L, where H = ⌊(n + 1)/2⌋ and L = ⌈(n + 1)/2⌉, the HOMO–LUMO gap is defined as as ΔG = μ H = μ L. In this article, a simple upper bound for the HOMO–LUMO gap corresponding to a special class of connected bipartite graphs is estimated. As an application, the upper bounds for the HOMO–LUMO gap of certain classes of nanotubes and nanotori are estimated.

Let A and B be n × n positive semidefinite matrices, f : [ 0 , ∞ ) → [ 0 , ∞ ) be an increasing convex function with f ( 0 ) = 0 , and let Φ : M n ( C ) → M n ( C ) be a positive unital linear map. It is shown that ∏ j = 1 k s j ( f 4 ( Φ ( A + B 2 ) ) ) ≤ 2 − 4 ∏ j = 1 k ( s j ( Φ 2 ( f 2 ( A ) ) + Φ ( f 2 ( A ) ) ) + ‖ Φ ( f 2 ( B ) ) ‖ + 1 ) × ( s j ( Φ 2 ( f 2 ( B ) ) + Φ ( f 2 ( B ) ) ) + ‖ Φ ( f 2 ( A ) ) ‖ + 1 ) for k = 1 , 2 , … , n , where s j ( T ) and ‖ T ‖ are the j th singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given. [ABSTRACT FROM AUTHOR]

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees. [ABSTRACT FROM AUTHOR]

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

Let Hn be the real space of nxn complex Hermitian matrices. Complete descriptions are given of the maps of Hn leaving invariant the pseudo spectral radius or the condition spectral radius of Jordan product of matrices. As application, maps on Hn that preserve the condition spectrum of Jordan product of matrices are classified. [ABSTRACT FROM AUTHOR]

Let λ j and a j j , 1 ≤ j ≤ n , be the eigenvalues and the diagonal entries of a Hermitian matrix A , both enumerated in the increasing order. We prove some inequalities that are stronger than the Schur majorisation inequalities ∑ j = 1 r λ j ≤ ∑ j = 1 r a j j , 1 ≤ r ≤ n. [ABSTRACT FROM AUTHOR]

We study real bihomogeneous polynomials r(z,\bar {z}) in n complex variables for which r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 is the squared norm of a holomorphic polynomial mapping. Such polynomials are the focus of the Sum of Squares Conjecture, which describes the possible ranks for the squared norm r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 and has important implications for the study of proper holomorphic mappings between balls in complex Euclidean spaces of different dimension. Questions about the possible signatures for r(z,\bar {z}) and the rank of r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of r. [ABSTRACT FROM AUTHOR]

A mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. Let α be a primitive nth root of unity, then the α−Hermitian adjacency matrix of a mixed graph is defined to be the matrix Hα = [hrs] where hrs = α if rs is an arc in D, hrs = α if sr is an arc in D, hrs = 1 if sr is a digon in D and hrs = 0 otherwise. In this paper we study the cospectrality of the Hermitian adjacency matrix of a mixed graph. [ABSTRACT FROM AUTHOR]

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees.

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

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 determine the underlying graphs of the mixed graphs with smallest Hermitian eigenvalue greater than − 3. Furthermore, we characterize all mixed graphs on n ≥ 11 vertices with smallest Hermitian eigenvalue greater than − 3. The mixed graphs on n ≤ 10 vertices with smallest Hermitian eigenvalue greater than − 3 could be also obtained easily with the help of computer because their underlying graphs are determined. Roughly speaking, we completely determine the mixed graphs with smallest Hermitian eigenvalue greater than − 3. [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]

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]

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]

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 receiver's architecture is always a critical issue in multiple-input multiple-output (MIMO) systems, as it has a great impact on the system's performance and complexity. Accordingly, in this paper, a spectral decomposition-based minimum mean squared error (MMSE) receiver is proposed in an uplink multi-user MIMO (MU-MIMO) system with a full-duplex decode and forward relay. Imperfect channel state information is assumed at the relay and base station where, least-squares channel estimation technique is applied due to its simplicity and low complexity. An equivalent relay is applied for self-interference cancellation. While, multi-user interference is mitigated by using the block diagonalization concept. A closed-form expression for the mean squared error (MSE) of the estimated data symbol is derived in the presence of channel estimation error. Moreover, an optimization problem is solved to derive an expression for the proposed MMSE receiver's matrix. Finally, the performance of the proposed receiver is compared to that of the conventional receiver and Zero-forcing (ZF) receiver. Results show the superiority of the proposed receiver over both receivers. [ABSTRACT FROM AUTHOR]

Let A be an n × n complex matrix. The spread of the matrix A, denotes by sp(A), is the largest distance between any two eigenvalues of A. It was introduced by L. Mirsky. In the present work we give new lower and upper bounds for the spread of A. Some of these bounds are explicit functions of the entries of A, others involve the rows and the columns of A. Bounds for the spread of the power of A are obtained too. Block matrices arise in many aspects of matrix theory, we present some bounds for the spread of the general 2 × 2 block matrix. [ABSTRACT FROM AUTHOR]

We investigate the change in the multiplicities of an eigenvalue of an Hermitian matrix whose graph is a general undirected graph, when an edge is removed from the graph. The same question when the graph is a tree has been investigated in prior work. Here, we give possible classifications of edges in a general graph in terms of the statuses of adjacent vertices. It turns out that there are four cases that do occur in a general graph but cannot occur in a tree. [ABSTRACT FROM AUTHOR]

In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A (k) converge to zero. However, this fact alone does not necessarily guarantee that A (k) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q (k). We prove that under certain assumptions A (k) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q (k) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q (k) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example. [ABSTRACT FROM AUTHOR]

In control engineering, numerical stability and real time are the two most important requirements for the matrix inversion. This article presents an efficient and robust method for the field-programmable gate array (FPGA) calculation of the matrix inversion. We initially consider the scenario that the matrix to be processed is a nonsingular Hermitian matrix. The proposed computation procedures are composed of the matrix decomposition, triangular matrix inversion, and matrices multiplication. The first procedure is completed by LDL factorization based on the outer form of Cholesky's method, whereas the recursive algorithm for block submatrices is adopted to achieve the triangular matrix inversion. The new method has the high level in the parallel pipelining mechanism and steals the characteristics of both the upper triangular matrix and its inversion to reduce the computation load and improve the numerical stability. Furthermore, the non-Hermitian matrix inversion can be easily solved if another procedure is added in the new method. Finally, we compare our method with the exiting FPGA-based techniques on one Xilinx Virtex-7 XC7VX690T FPGA. Meanwhile, it has solved one array antenna control problem of the adaptive digital beam forming for one phased array radar successfully. [ABSTRACT FROM AUTHOR]

Abstract 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.