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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]

In many stability problems, the investigation of pure imaginary roots for a polynomial family is very important. Given a pure imaginary complex number, the set of all images of uncertainty vectors is called the value set corresponding to this pure imaginary complex number. The question whether these sets contain the origin is very important from robust stability point of view of a polynomial family. Cutoff frequency guarantees the noninclusion of the origin to the value set for large frequencies. In this paper, we give a procedure for more strict estimation of cutoff frequency and applications of the obtained result to the constant inertia problem of a polynomial family. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be a fixed integer, and denote by M n the algebra of all n × n complex matrices. Fix also an integer k such that 1 ≤ k < n. We prove that if φ : M n → M n is a linear map which preserves the polynomial numerical radius of order k , there exist then a unitary n × n complex matrix U and a complex number ξ of modulus one such that either φ (T) = ξ U ⁎ T U for all T ∈ M n , or φ (T) = ξ U ⁎ T t U for all T ∈ M n. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the solution of the autonomous q -difference system D q Y x = A Y q x with the initial condition Y 0 = Y 0 where A is a constant square complex matrix, D q is the Jackson q -derivative and 0 < q < 1 , is asymptotically stable if and only if ℜ λ < 0 for all λ ∈ σ A where σ A is the set of all eigenvalues of A (the spectrum of A). This results are exploited to provide the orthogonality property of the discrete q -Hermite matrix polynomials. [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]

Since the non-commutativity and particular structure of the quaternion algebra, the quternion difference equations (short for QDCEs) have a large difference from the classical theory of difference equations. In this paper, we establish a general theory of the higher-order linear QDCEs including the criteria of the linear independence (or dependence) of the discrete functions in the quaternion space, Liouville formulas, the structure theorems of the general solutions and the particular solutions with the quaternion power and trigonometric form, etc. By introducing the complex adjoint difference equations and the quaternion characteristic polynomial of the higher-order linear QDCEs, some basic results of the homogeneous and non-homogeneous difference equations are obtained. Through the analysis of the complex adjoint matrix and the quaternion eigenvalue, the general solutions of the higher-order linear QDCEs with variable and with constant coefficients are established. Several methods of obtaining the general solutions for the higher-order linear QDCEs are demonstrated and several examples are provided to illustrate the feasibility of our obtained results. [ABSTRACT FROM AUTHOR]

We determine the generic complete eigenstructures for n \times n complex symmetric matrix polynomials of odd grade d and rank at most r. More precisely, we show that the set of n Ã-- n complex symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and rank at most r is the union of the closures of the [rd/2] + 1 sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then we prove that these sets are open in the set of nÃ--n complex symmetric matrix polynomials of odd grade d and rank at most r. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure in the case where all their elementary divisors are different from each other and of degree 1. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than 1, because the symmetric ones include eigenvalues while the others not. This difference requires using new techniques. [ABSTRACT FROM AUTHOR]

The zeros of the random Laurent series 1 / μ - ∑ j = 1 ∞ c j / z j , where each c j is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement z ↦ 1 / z , we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the correlation functions for the random Maclaurin series can be determined. Only for | μ | → ∞ is a determinantal point process obtained. For the one and two point correlations, by regarding the Maclaurin series as the limit of a random polynomial, a direct calculation can also be given. [ABSTRACT FROM AUTHOR]