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Classical matrix theory for real, complex and hypercomplex numbers is a well-known concept. Is it possible to construct matrix theory over dual-generalized complex (DGC) matrices? The answer to this question is given in this paper. The paper is constructed as follows. Firstly, the fundamental concepts for DGC matrices are introduced and DGC special matrices are defined. Then, theoretical results related to eigenvalues/eigenvectors are obtained and universal similarity factorization equality (USFE) regarding to the dual fundamental matrix are presented. Also, spectral theorems for Hermitian and unitary matrices are introduced. Finally, due to the importance of unitary matrices, a method for finding a DGC unitary matrix is stated and examples for spectral theorem are given. [ABSTRACT FROM AUTHOR]

The problem for the Bessel equation of an integer order with complex physical and spectral parameters in the boundary condition is considered. The spectral parameter enters the boundary condition quadratically. The question of the basis property of the system of eigenfunctions in the case of the appearance of a multiple eigenvalue is studied. [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same. [ABSTRACT FROM AUTHOR]

Since the eigenfrequency of gravitational waves from cold neutron stars becomes a complex number, where the real and imaginary parts respectively correspond to an oscillation frequency and damping rate, one has to somehow solve the eigenvalue problem concerning the eigenvalue in two-dimensional parameter space. To avoid this bother, one sometimes adopts an approximation, where the eigenvalue is in one-dimensional parameter space. In this study, first, we show the accuracy of the zero-damping approximation, which is one of the one-dimensional approximations, for the fundamental and 1st pressure modes. But, this approximation is not applicable to the spacetime mode, because the damping rate of the spacetime mode is generally comparable to the oscillation frequency. Nevertheless, we find the empirical relation for the ratio of the imaginary part to the real part of the eigenfrequency, which is expressed as a function of the steller compactness almost independently of the adopted equations of state for neutron star matter. Adopting this empirical relation, one can express the eigenfrequency in terms of just the real part, i.e., the problem to solve becomes an eigenvalue problem with a one-dimensional eigenvalue. Then, we find that the frequencies are estimated with good accuracy even with such approximations even for the 1st spacetime mode. [ABSTRACT FROM AUTHOR]

Let μ 1 be a complex number in the numerical range W (A) of a normal matrix A. In the case when no eigenvalues of A lie in the interior of W (A) , we identify the smallest convex region containing all possible complex numbers μ 2 for which μ 1 ∗ 0 μ 2 is a 2-by-2 compression of A. [ABSTRACT FROM AUTHOR]

A complex number µ is called an extended eigenvalue for an operator T ∈ B(H) on a Hilbert space H if there exists a nonzero operator T ∈B(H)such that: TX = µXT, such X is called an extended eigenoperator corresponding to. The goal of this paper is to calculate extended eigenvalues and extended eigenoperators for the weighted unilateral (Forward and Backward) shift operators. We also find an extended eigenvalues for weighted bilateral shift operator. Moreover, the closed ness of extended eigenvalues for the weighted unilateral (Forward and Backward) shift operators under multiplication is proven. [ABSTRACT FROM AUTHOR]

We investigate the Sturm-Liouville (S-L) operator with boundary and transfer conditions dependent on the eigen-parameter. By utilizing interval partitioning and factorization techniques of characteristic function, it is proven that this problem has a finite number of eigenvalues when the coefficients of the equation meet certain conditions, and some conditions for determining the number of eigenvalues are provided. The results indicate that the number of eigenvalues in this problem varies when the transfer conditions depend on the eigen-parameter. Furthermore, the equivalence between this problem and matrix eigenvalue problems is studied, and an equivalent matrix representation of the S-L problem is presented. [ABSTRACT FROM AUTHOR]

The eigenvalues of complex q-Sturm–Liouville boundary value problems are the focus of this paper. The coefficients of the corresponding q-Sturm–Liouville equation provide the lower bounds on the real parts of all eigenvalues, and the real part of the eigenvalue and the coefficients of this q-Sturm–Liouville equation provide the bounds on the imaginary part of each eigenvalue. [ABSTRACT FROM AUTHOR]

Based on the novel prescription for the power function, (x + i y) m , the new expression for Ψ (x , y | m) , the eigenfunction of the operator of the third component of the angular momentum, M ^ z , is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x + i y) m = ρ m e i m ϕ , are invariant under the rotations at 2 π for any, not necessarily integer, m—the eigenvalue of M ^ z . For any real m the functions Ψ (x , y | m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M ^ z . The eigenfunctions and eigenvalues of the angular momentum operator squared, M ^ 2 , derived for the two different prescriptions for the square root, (m 2) 1 / 2 , (m 2) 1 / 2 = | m | and (m 2) 1 / 2 = ± m , are reported. The normalizable eigenfunctions of M ^ 2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M ^ 2 . [ABSTRACT FROM AUTHOR]

In this paper, we propose and solve low phase-rank approximation problems, which serve as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. It is well known that a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix being defined as the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with the matrix arithmetic mean, it turns out that a natural parallel for the low phase-rank approximation problem is to use the matrix geometric mean to measure the approximation error. Importantly, we derive a majorization inequality between the phases of the geometric mean and the arithmetic mean of the phases, similarly to the Ky-Fan inequality for eigenvalues of Hermitian matrices. A characterization of the solutions to the proposed problem, with the same flavor as the Schmidt-Mirsky theorem, is then obtained in the case where both the objective matrix and the approximant are restricted to be positive-imaginary. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary. [ABSTRACT FROM AUTHOR]

A novel method for finding the eigenvalues of a Sturm–Liouville problem is developed. Following the minimalist approach, the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although the resulting equation is nonlinear, its form allows us to find the general solution by adding a second part to a particular solution. This splitting of the general solution into two parts involves the Schwarzian derivative: hence, the name of the approach. The eigenvalues that correspond to acceptable solutions can be found by requiring the second part to correct the asymptotically diverging behavior of the particular solution. The method can be applied to many different areas of physics, such as the Schrödinger equation in quantum mechanics and stability problems in fluid dynamics. Examples are presented. [ABSTRACT FROM AUTHOR]

We report a study of the entanglement between the quantized photon field and an atom arising in the photo-ionization process. Our approach is based on an ab initio solution of the time-dependent Schrödinger equation (TDSE) describing the quantum evolution of a bipartite system consisting of the atom and the quantized electromagnetic field. Using the solution of the TDSE, we calculate the reduced photon density matrix, which we subsequently use to compute entanglement entropy. We explain some properties of the entanglement entropy and propose an approximate formula for the entanglement entropy based on the analysis of the density matrix and its eigenvalues. We present the results of a comparative study of the entanglement in the photo-ionization process for various ionization regimes, including the tunneling and the multiphoton ionization regimes. [ABSTRACT FROM AUTHOR]

The concept of $ C_{\pi }^R $ C π R -matrix is extended to $ C_{\pi }^R $ C π R -tensor, which also generalizes the concept of C-tensor. A necessary and sufficient condition for a tensor to be a $ C_{\pi }^R $ C π R -tensor is provided. We analyse decompositions of $ C_{\pi }^R $ C π R -tensors and prove that $ C_{\pi }^R $ C π R -tensors are nonsingular. Positive linear combinations and Hadamard product of two $ C_{\pi }^R $ C π R -tensors are also discussed. Finally, some properties of $ B_{\pi }^R $ B π R -tensor are given to localize real eigenvalues of a tensor. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the p -Laplacian for any p ∈ (1 , + ∞) (this paper covers the case p = 1 whereas the case p = + ∞ was already known). [ABSTRACT FROM AUTHOR]

We consider Hamiltonians of models describing non-relativistic quantum mechanical matter coupled to a relativistic field of bosons. If the free Hamiltonian has an eigenvalue, we show that this eigenvalue persists also for nonzero coupling. The eigenvalue of the free Hamiltonian may be degenerate provided there exists a symmetry group acting irreducibly on the eigenspace. Furthermore, if the Hamiltonian depends analytically on external parameters then so does the eigenvalue and eigenvector. Our result applies to the ground state as well as resonance states. For our results, we assume a mild infrared condition. The proof is based on operator theoretic renormalization. It generalizes the method used in Griesemer and Hasler (Ann Henri Poincaré 10(3):577–621, 2009) to non-degenerate situations, where the degeneracy is protected by a symmetry group, and utilizes Schur's lemma from representation theory. [ABSTRACT FROM AUTHOR]

Seniority-zero geminal wavefunctions are known to capture bond-breaking correlation. Among this class of wavefunctions, Richardson–Gaudin states stand out as they are eigenvectors of a model Hamiltonian. This provides a clear physical picture, clean expressions for reduced density matrix (RDM) elements, and systematic improvement (with a complete set of eigenvectors). Known expressions for the RDM elements require the computation of rapidities, which are obtained by first solving for the so-called eigenvalue based variables (EBV) and then root-finding a Lagrange interpolation polynomial. In this paper, we obtain expressions for the RDM elements directly in terms of the EBV. The final expressions can be computed at the same cost as the rapidity expressions. Therefore, except, in particular, circumstances, it is entirely unnecessary to compute rapidities at all. The RDM elements require numerically inverting a matrix, and while this is usually undesirable, we demonstrate that it is stable, except when there is degeneracy in the single-particle energies. In such cases, a different construction would be required. [ABSTRACT FROM AUTHOR]

We give real representations of $ A^n $ A n (or $ e^{t A} $ e tA ) based on $ A^nP_{\mu } $ A n P μ for a real square matrix A, where $ P_{\mu } $ P μ is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients. [ABSTRACT FROM AUTHOR]

In this paper we prove that every totally real algebraic integer λ of degree d ≥ 2 occurs as an eigenvalue of some tree of height at most d (d + 1) / 2 + 3 . In order to prove this, for a given algebraic number α ≠ 0 , we investigate an additive semigroup that contains zero and is closed under the map x ↦ α / (1 - x) for x ≠ 1 . The problem of finding the smallest such semigroup seems to be of independent interest. [ABSTRACT FROM AUTHOR]

Let A be a bounded linear operator in a complex separable Hilbert space, A* be its adjoint one and AI := (A - A*)/(2i). Assuming that AI is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of A. Our results are formulated in terms of the "extended" eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality P 8k =1(Im k(A))2 = N2 2 (AI), where k(A) (k = 1, 2, . . .) are the eigenvalues of A and N2(·) is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators. [ABSTRACT FROM AUTHOR]

This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Liouville problem are transformed into an eigenvalue problem of a certain self-adjoint operator. Next, the eigenfunctions of the problem and some properties of the eigenvalues are given via construction of the basic solution. The Green's function for the Sturm-Liouville problem is also given. Finally, the continuity of the eigenvalues and eigenfunctions of the problem is discussed. Especially, the differential expressions of the eigenvalues for some parameters have been obtained, including the parameters in the eigenparameter-dependent transmission conditions. [ABSTRACT FROM AUTHOR]

Non‐Hermitian Hamiltonians may still possess real eigenvalues in case of the existence of parity‐time (PT) symmetry. Exceptional points (EPs) occur at the phase transition from real to complex eigenvalues due to PT‐symmetry breaking in the parameter space. Magnonic devices use magnons to carry, transport, and process information, which have the advantages of low energy dissipation, wave‐based computing, and nonlinear data processing. The combination of PT‐symmetry and magnonics may lead to novel physics as well as unprecedented functional device applications. Recently, the research of PT‐symmetry in magnetism has developed rapidly. In this review, the theoretical predictions as well as experimental findings of PT‐symmetry in magnetism are summarized. First, a brief introduction to PT symmetry, EPs, and anti‐PT symmetry is presented. Second, the theoretical and experimental progress of magnonic PT symmetry are summarized. Third, the theoretical predictions of higher‐order EPs and anti‐PT symmetry in magnonic systems are given. Finally, the study concludes by discussing the future challenges and research prospects in magnonic PT‐symmetry, and proposals for experimental observations of magnonic higher‐order EPs and anti‐PT symmetry are suggested. [ABSTRACT FROM AUTHOR]

The spectrum situation awareness problem in space–air–ground integrated networks (SAGINs) is studied from a tensor-computing perspective. Tensor and tensor computing, including tensor decomposition, tensor completion and tensor eigenvalues, can satisfy the application requirements of SAGINs. Tensors can effectively handle multidimensional heterogeneous big data generated by SAGINs. Tensor computing is used to process the big data, with tensor decomposition being used for dimensionality reduction to reduce storage space, and tensor completion utilized for numeric supplementation to overcome the missing data problem. Notably, tensor eigenvalues are used to indicate the intrinsic correlations within the big data. A tensor data model is designed for space–air–ground integrated networks from multiple dimensions. Based on the multidimensional tensor data model, a novel tensor-computing-based spectrum situation awareness scheme is proposed. Two tensor eigenvalue calculation algorithms are studied to generate tensor eigenvalues. The distribution characteristics of tensor eigenvalues are used to design spectrum sensing schemes with hypothesis tests. The main advantage of this algorithm based on tensor eigenvalue distributions is that the statistics of spectrum situation awareness can be completely characterized by tensor eigenvalues. The feasibility of spectrum situation awareness based on tensor eigenvalues is evaluated by simulation results. The new application paradigm of tensor eigenvalue provides a novel direction for practical applications of tensor theory. [ABSTRACT FROM AUTHOR]

A novel unconstrained optimization model named weighted trace-penalty minimization (WTPM) is proposed to address the extreme eigenvalue problem arising from the full configuration interaction (FCI) method. Theoretical analysis shows that the global minimizers of the WTPM objective function are the desired eigenvectors, rather than the eigenspace. Analyzing the condition number of the Hessian operator in detail contributes to the determination of a near-optimal weight matrix. With the sparse feature of FCI matrices in mind, the coordinate descent (CD) method is adapted to WTPM and results in the WTPM-CD method. The reduction of computational and storage costs in each iteration shows the efficiency of the proposed algorithm. Finally, the numerical experiments demonstrate the capability to address large-scale FCI matrices. [ABSTRACT FROM AUTHOR]

We investigate a three-component system involving the Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Our goal is to investigate the connection between homoclinic snaking and semi-strength interaction in a three-variable reaction–diffusion system. A two-parameter bifurcation diagram of homogeneous, periodic and localized patterns is obtained numerically, and a natural asymptotic scaling for semi-strong interaction theory is found where an activator source term a=O(δ1) and b=O(δ1) , with δ1≪1 being the diffusion ratio. Under this regime, singular perturbation techniques are used to construct localized steady-state patterns, and new types of non-local eigenvalue problems (NLEP) are derived to determine the stability of these patterns to O(1) time-scale instabilities. We extend the scope of the NLEP by considering a general scenario where both time-scaling parameters are non-zero. All analytical results are found to agree with numerics. Further numerical results are presented on the location of various types of breathing Hopf instability for localized patterns. [ABSTRACT FROM AUTHOR]

The evaluation of leading industries involves the use of evaluation index data that exhibit multi-source uncertainty. This implies that the evaluation index values encompass various types of numbers, such as grey numbers, white numbers, fuzzy numbers, interval value fuzzy numbers, and more. The evaluation index system possesses a multi-level structure with interconnections among the indicators. In this study, we propose a generalized grey relational weighted evaluation model based on multi-source uncertainty for assessing leading industries. To begin with, the generalized grey number is used to represent all uncertain data, and the weights of multi-level indicators are calculated by combining the generalized grey number with the information entropy model. Subsequently, the correlation between the indexes is examined utilizing the theorem of the generalized grey absolute relational degree model. This analysis leads to the construction of a generalized grey absolute relational degree matrix, from which eigenvalues and eigenvectors are derived. Based on the generalized grey weight of the multi-level indexes, the eigenvalue and eigenvector of the generalized grey absolute relational matrix, and the grey relational weighted evaluation model for leading industry evaluation are established. Finally, the effectiveness and feasibility of the proposed model are validated through a case study involving the evaluation of the leading industry in the airport economic zone. [ABSTRACT FROM AUTHOR]

We obtain the generic complete eigenstructures of complex Hermitian n times n matrix pencils with rank at most r (with r leq n). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian n times n pencils with the same complete eigenstructure (up to the specific values of the distinct finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases r =n, corresponding to general Hermitian pencils, and r < n exhibit surprising differences, since for r < n the generic complete eigenstructures can contain only real eigenvalues, while for r = n they can contain real and nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures. [ABSTRACT FROM AUTHOR]

It has long been believed that skin modes are equivalent to the nontrivial point gap. However, we find that this concomitance can be broken, in that skin modes can be absent or present when the point gap is nontrivial or trivial, respectively, named anomalous non-Hermitian skin effect. This anomalous phenomenon arises whenever unidirectional hopping amplitudes emerge among subsystems, where sub-chains have decoupling-like behaviors and contribute only to the energy levels without particle occupation. The occurrence of anomalous non-Hermitian skin effect is accompanied by changes in open boundary eigenvalues, whose structure exhibits multifold exceptional points and can not be recovered by continuum bands. Moreover, an experimental setup is proposed to simulate this effect. Our results reveal the topologically inequivalence of skin modes and point gap. This effect not only provides a deeper understanding of non-Bloch theory and critical phenomena, but may inspire applications, such as in sensor field. Non-Hermitian skin effect, as an important consequence of a non-Hermitian topological system, has recently attracted great attention. This paper reports an anomalous non-Hermitian skin effect, where the correspondence between skin modes and the non-zero winding number defined within point gaps can be broken, enabling further understanding of the non-Bloch band theory in the non-Hermitian field. [ABSTRACT FROM AUTHOR]

The current study is intended to provide a comprehensive application of Sturm-Liouville (S-L) problem by benefiting from the proportional derivative which is a crucial mathematical tool in control theory. This advantageous derivative, which has been presented to the literature with an interesting approach and a strong theoretical background, is defined by two tuning parameters in control theory and a proportional-derivative controller. Accordingly, this research is presented mainly to introduce the beneficial properties of the proportional derivative for analyzing the S-L initial value problem. In addition, we reach a novel representation of solutions for the S-L problem having an importing place in physics, supported by various graphs including different values of arbitrary order and eigenvalues under a specific potential function. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate some inequalities for positive linear maps on matrices. Moreover, we discuss lower bounds for the spread of nonnegative matrices which provides improvement over the existing bounds. [ABSTRACT FROM AUTHOR]

The eigenstructure of imaginary axis eigenvalues of a Hamiltonian matrix is of importance in many fields of control systems, for example, in stability analysis of linear Hamiltonian systems, computation of the solutions of an algebraic Riccati equation (ARE), existence of a Lyapunov function for LTI systems, etc. The dynamical system consisting of all stationary trajectories for an optimal control problem – often called the Hamiltonian system – is known to admit, in its state space dynamical equation, a Hamiltonian matrix for the system matrix. In each of these cases, defectiveness of imaginary axis eigenvalues of a Hamiltonian matrix turns out to be of crucial importance. For example, defectiveness causes unboundedness of the oscillatory stationary trajectories in the Hamiltonian system. A characterisation of the ARE solutions in terms of Lagrangian invariant subspaces of the Hamiltonian matrix when the imaginary axis eigenvalues are defective has been addressed in the literature for the controllable case. This paper focuses on the general case of uncontrollable systems; we formulate conditions under which the imaginary axis eigenvalues of the Hamiltonian matrix are non-defective: this is central for solutions corresponding to imaginary axis eigenvalues to not become unbounded. We provide conditions on the so-called ϵ -characteristic of the non-defective imaginary axis eigenvalue: this helps in the characterisation of Lagrangian invariant subspaces. We formulate conditions under which a passivity-based Hamiltonian matrix is normal (i.e. it commutes with its transpose), and we link normality of such Hamiltonian matrices with all-pass behaviour. In summary, this paper formulates results that link defectiveness of imaginary axis eigenvalues of Hamiltonian matrix to solvabilities of Lyapunov and Algebraic Riccati equations, controllability/observability, ϵ -characteristic and sign-controllability. We consider examples in the area of bounded-real transfer functions and RLC circuits, both controllable and uncontrollable, to study applicability of the results of this paper. [ABSTRACT FROM AUTHOR]

In order to further connect structured matrices and integer sequences, r -circulant matrices involving the generalized Narayana numbers are considered. Estimates for spectral norms bounds of such matrices are presented and their eigenvalues are determined. Moreover, the conditions under which the circulant matrix and the skew circulant matrix involving generalized Narayana numbers are invertible are given. In particular, it is shown that every circulant matrix with Narayana numbers is necessarily invertible. [ABSTRACT FROM AUTHOR]

The Schrödinger equation with a quartic potential is a fundamental equation describing the physics of stimulated side scattering, but no exact, even approximate solution has been formulated yet. Here, the approximate solutions to such a Schrödinger equation are derived using the WKBJ method under different variables. The solutions are accurate enough in most cases. Eigenvalues and eigenfunctions, which are the most impressive properties of such equations, are also discussed. [ABSTRACT FROM AUTHOR]

Equivariant Pyragas control is a delayed feedback method that aims to stabilize spatio-temporal patterns in systems with symmetries. In this article, we apply equivariant Pyragas control to discrete waves, which are periodic solutions that have a finite number of spatio-temporal symmetries. We prove sufficient conditions under which a discrete wave can be stabilized via equivariant Pyragas control. The result is applicable to a broad class of discrete waves, including discrete waves that are far away from a bifurcation point. Key ingredients of the proof are an adaptation of Floquet theory to systems with symmetries, and the use of characteristic matrix functions to reduce the infinite dimensional eigenvalue problem to a one dimensional zero finding problem. [ABSTRACT FROM AUTHOR]

We study the matrix completion problem: an underlying m × n matrix P is low rank, with incoherent singular vectors, and a random m × n matrix A is equal to P on a (uniformly) random subset of entries of size dn. All other entries of A are equal to zero. The goal is to retrieve information on P from the observation of A. Let A 1 be the random matrix where each entry of A is multiplied by an independent { 0 , 1 } -Bernoulli random variable with parameter 1/2. This paper is about when, how and why the non-Hermitian eigen-spectra of the matrices A 1 (A - A 1) ∗ and (A - A 1) ∗ A 1 captures more of the relevant information about the principal component structure of A than the eigen-spectra of A A ∗ and A ∗ A . We show that the eigenvalues of the asymmetric matrices A 1 (A - A 1) ∗ and (A - A 1) ∗ A 1 with modulus greater than a detection threshold are asymptotically equal to the eigenvalues of P P ∗ and P ∗ P and that the associated eigenvectors are aligned as well. The central surprise is that by intentionally inducing asymmetry and additional randomness via the A 1 matrix, we can extract more information than if we had worked with the singular value decomposition (SVD) of A. The associated detection threshold is asymptotically exact and is non-universal since it explicitly depends on the element-wise distribution of the underlying matrix P. We show that reliable, statistically optimal but not perfect matrix recovery, via a universal data-driven algorithm, is possible above this detection threshold using the information extracted from the asymmetric eigen-decompositions. Averaging the left and right eigenvectors provably improves estimation accuracy but not the detection threshold. Our results encompass the very sparse regime where d is of order 1 where matrix completion via the SVD of A fails or produces unreliable recovery. We define another variant of this asymmetric principal component analysis procedure that bypasses the randomization step and has a detection threshold that is smaller by a constant factor but with a computational cost that is larger by a polynomial factor of the number of observed entries. Both detection thresholds allow to go beyond the barrier due to the well-known information theoretical limit d ≍ log n for exact matrix completion found in the literature. [ABSTRACT FROM AUTHOR]

We consider an operator of multiplication by a complex-valued potential in L 2 (R) , to which we add a convolution operator multiplied by a small parameter. The convolution kernel is supposed to be an element of L 1 (R) , while the potential is a Fourier image of some function from the same space. The considered operator is not supposed to be self-adjoint. We find the essential spectrum of such an operator in an explicit form. We show that the entire spectrum is located in a thin neighbourhood of the spectrum of the multiplication operator. Our main result states that in some fixed neighbourhood of a typical part of the spectrum of the non-perturbed operator, there are no eigenvalues and no points of the residual spectrum of the perturbed one. As a consequence, we conclude that the point and residual spectrum can emerge only in vicinities of certain thresholds in the spectrum of the non-perturbed operator. We also provide simple sufficient conditions ensuring that the considered operator has no residual spectrum at all. [ABSTRACT FROM AUTHOR]

We discuss some simple Hückel-like matrix representations of non-Hermitian operators with antiunitary symmetries that include generalized P T (parity transformation followed by time-reversal) symmetry. One of them exhibits extremely broken antiunitary symmetry (complex eigenvalues for all nontrivial values of the model parameter) because of the degeneracy of the operator in the Hermitian limit. These examples illustrate the effect of point-group symmetry on the spectrum of the non-Hermitian operators. We construct the necessary unitary matrices by means of simple graphical representations of the non-Hermitian operators. [ABSTRACT FROM AUTHOR]

Further elaborations on the adjugate method for eigenvalue–eigenvector assignment are discussed in this paper. Additional results concerning the admissible pair introduced recently are stated, verified, and commented on. Properties of the adjugate method concerning eigenvalue reassignment of controllable and uncontrollable systems are revealed and pursued further in this study A reformulation of how calculations should be carried out is investigated, pointing out merits, bounds, and limitations. It has been found that reassignment can involve a case where the closed-loop eigenvector companion z i is z i = 0 and the associated eligible closed-loop eigenvectors w i are open-loop ones. This can be considered an advantage in the sense of knowing z i and w i beforehand. This also applies to the case of uncontrollable eigenvalue reassignment, where enlarged closed-loop eigenvector subspaces are uncovered, enabling more flexible designs. Assessments of the traditional method compared to the adjugate method are commented on whenever appropriate. Numerical issues are pointed out where Leverrier's algorithm is adapted for efficient matrix null space and matrix adjugate determination. A feature of the adjugate method is a result where w U = 0 , indicating that the eigenvalue assigned is uncontrollable. Such a distinctive feature enables a classification procedure for systems regarding controllability and observability. The various concepts pointed out have been demonstrated and authenticated through carefully selected examples. [ABSTRACT FROM AUTHOR]

Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two alternative methods. The first technique is based on a projection onto subspaces with dimension equal to the normal rank of the pencil while the second approach exploits an augmented matrix pencil. The projection approach seems to be the most attractive version for generic singular pencils because of its efficiency, while the augmented pencil approach may be suitable for applications where a linear system with the augmented pencil can be solved efficiently. [ABSTRACT FROM AUTHOR]

We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices. [ABSTRACT FROM AUTHOR]

Stenger conjectures are claims about the location of the eigenvalues of matrices whose elements are certain integrals involving basic Lagrange interpolating polynomials supported on the zeros of orthogonal polynomials. In this paper, we show the validity of the extended Stenger conjecture for families of classical orthogonal polynomials. We also show the validity of the restricted Strenger conjecture for a family of Jacobi and generalized Laguerre orthogonal polynomials. A connection with the A-stability of the collocation Runge-Kutta methods is investigated. [ABSTRACT FROM AUTHOR]

Summary: We study cooperative output feedback tracking control of stochastic nonlinear heterogeneous leader‐following multi‐agent systems. Each agent has a continuous‐time stochastic nonlinear heterogeneous dynamics with an unmeasurable state, and there are additive and multiplicative noises along with information exchange among agents. We propose admissible distributed observation strategies for estimating the leader's and the followers' states, and admissible cooperative output feedback control strategies based on the certainty equivalence principle. By output regulation theory and stochastic analysis, we show if the dynamics of each agent satisfies the Lipschitz condition, and the product of the leader's Lipschitz coefficient, the intensity of multiplicative measurement noises, and the constant related to the leader's Lipschitz coefficient and dimension is less than 1/4$$ 1/4 $$ the minimum nonzero eigenvalue of graph Laplacian, then there exist admissible distributed observation and cooperative control strategies to ensure mean square bounded output tracking. Finally, the effectiveness of our control strategies is demonstrated by a numerical simulation. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices Λ = diag { λ 1 , ... , λ p } ∈ C p × p and X = [ x 1 , ... , x p ] ∈ C n × p with p ≤ n , λ i ≠ λ j for i ≠ j , i , j = 1 , ... , p , rank (X) = p and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that M X Λ 2 + D X Λ + K X = 0. Problem OAP: Given real symmetric matrices M a , D a and K a , find (M ^ , D ^ , K ^) ∈ S E such that ‖ M ^ − M a ‖ N 2 + ‖ D ^ − D a ‖ N 2 + ‖ K ^ − K a ‖ N 2 = min (M , D , K) ∈ S E (‖ M − M a ‖ N 2 + ‖ D − D a ‖ N 2 + ‖ K − K a ‖ N 2) , where S E is the solution set of Problem IQEP and ‖ ⋅ ‖ N is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data. [ABSTRACT FROM AUTHOR]

We establish an asymptotic formula for the eigenvalues of a 2 × 2 -system of ordinary differential equations provided with separated boundary conditions when the coefficients are (possibly singular) continuous, real, and finite measures. [ABSTRACT FROM AUTHOR]

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eigenvalues. Furthermore, we prove that the proposed algorithms converge, and linear convergence is also proved. Numerical examples are presented to demonstrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

A subset Y of the general linear group $\text{GL}(n,q)$ is called t -intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t -dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$. We show that, if n is sufficiently large compared to t , the size of every such t -intersecting set is at most that of the stabiliser of a basis of a t -dimensional subspace of $\mathbb{F}_q^n$. In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t -dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$. These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q -analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$. [ABSTRACT FROM AUTHOR]

We study the spectral stability of the nonlinear Dirac operator in dimension 1 + 1 , restricting our attention to nonlinearities of the form f (ψ , β ψ C 2) β . We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form e - i ω t ϕ 0 . For the case of power nonlinearities f (s) = s | s | p - 1 , p > 0 , we obtain a range of frequencies ω such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition ϕ 0 , β ϕ 0 C 2 > 0 characterizes groundstates analogously to the Schrödinger case. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyze a mixed method for linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise (k+1), k and (k+1)-th degree polynomial functions (k=1), respectively. The numerical eigenfunction of stress is symmetric. By the discrete H1-stability of numerical displacement, we prove an O(hk+2) approximation to the L²-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem, with proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an O(h²) initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete H1-stability of numerical displacement, while only an O(h) approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments. [ABSTRACT FROM AUTHOR]

We consider an overdetermined eigenvalue problem related to the MEMS operator given by Lτ := Δ2 − τΔ on a smooth bounded domain Ω ⊂ RN, N ≥ 2. We give radial solutions on balls. Moreover, we establish a symmetry result with respect to operator Lτ , that is, under some hypotheses, we show that if a solution does exist to the overdetermined eigenvalue problem, then the domain Ω must be a ball. [ABSTRACT FROM AUTHOR]

In this paper we introduce a different types of matrices. We determine the characteristic polynomial theoretically and by using software. In some cases we draw conclusions from it about the eigenvalues of the matrices. The proofs are sometimes constructive in the sense that we construct a basis of the nullspace. [ABSTRACT FROM AUTHOR]