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We begin a study of Schur analysis when the variable is now a matrix rather than a complex number. We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers. [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type I. In our framework, each theta relation is controlled by combinatorial properties of a pair (T , P) of a regular matrix T over an imaginary quadratic field and a positive-definite Hermitian matrix P over the complex number field. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine all matrix representations of the real numbers. It is shown that every such representation is completely reducible, while all non-trivial irreducible representations must be of 2-dimensional and can be expressed in a unique form. It is found that those representations are essentially determined by the ways of embedding the real numbers into the complex numbers. This results in a one-to-one correspondence between the equivalent classes of irreducible representations and the equivalent classes of homomorphisms from the real number field to the complex number field. The matrix representations of the complex numbers are also determined. [ABSTRACT FROM AUTHOR]

Consider a nonzero contraction T and a bounded operator X satisfying T X = q X T for a complex number q. There are some interesting results in the literature on q -commuting dilation and q -commutant lifting of such pair (T , X) when | q | = 1. Here we improve a few of them to the class of scalars q satisfying | q | ≤ 1 ‖ T ‖. [ABSTRACT FROM AUTHOR]

We describe transposed Poisson algebra structures on Block Lie algebras B (q) and Block Lie superalgebras S (q) , where q is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on B (q) are trivial whenever q ∉ Z , and for each q ∈ Z there is only one (up to an isomorphism) non-trivial transposed Poisson structure on B (q). The superalgebra S (q) admits only trivial transposed Poisson superalgebra structures for q ≠ 0 and two non-isomorphic non-trivial transposed Poisson superalgebra structures for q = 0. As a consequence, new Lie algebras and superalgebras that admit non-trivial Hom-Lie algebra structures are found. [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 complex unit gain graph (T -gain graph), Φ = (G , φ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G , and its inverse is assigned to the opposite orientation. A T -gain graph Φ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is 1. Signed graphs are special cases of T -gain graphs. The adjacency matrix of Φ, denoted by A (Φ) is defined canonically. The gain Laplacian for Φ is defined as L (Φ) = D (Φ) − A (Φ) , where D (Φ) is the diagonal matrix with diagonal entries are the degrees of the vertices of G. The minimum number of vertices (resp., edges) to be deleted from Φ in order to get a balanced gain graph is the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of L (Φ). We provide several lower and upper bounds for extremal eigenvalues of L (Φ) in terms of different graph parameters such as the number of edges, vertex degrees, and average 2-degrees. The signed graphs are particular cases of the T -gain graphs, all the bounds established in this paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erdős-Reýni graphs. Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds. [ABSTRACT FROM AUTHOR]

We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle. [ABSTRACT FROM AUTHOR]

A T -gain graph, Φ = (G , φ) , is a graph in which the function φ assigns a unit complex number to each orientation of an edge of G , and its inverse is assigned to the opposite orientation. The associated adjacency matrix A (Φ) is defined canonically. The energy E (Φ) of a T -gain graph Φ is the sum of the absolute values of all eigenvalues of A (Φ). We study the notion of energy of a vertex of a T -gain graph, and establish bounds for it. For any T -gain graph Φ, we prove that 2 τ (G) − 2 c (G) ≤ E (Φ) ≤ 2 τ (G) Δ (G) , where τ (G) , c (G) and Δ (G) are the vertex cover number, the number of odd cycles and the largest vertex degree of G , respectively. Furthermore, using the properties of vertex energy, we characterize the class of T -gain graphs for which E (Φ) = 2 τ (G) − 2 c (G) holds. Also, we characterize the T -gain graphs for which E (Φ) = 2 τ (G) Δ (G) holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix. [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]

The idea of decomposing a matrix into a product of structured matrices such as triangular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions over complex number field, extending our work in [5] . We prove that every n × n complex matrix is a product of finitely many tridiagonal, skew symmetric (when n is even), companion and generalized Vandermonde matrices, respectively. We also prove that a generic complex n × n centrosymmetric matrix is a product of finitely many symmetric Toeplitz (resp. persymmetric Hankel) matrices. We determine an upper bound of the number of structured matrices needed to decompose a matrix for each case. [ABSTRACT FROM AUTHOR]

We introduce the concept of general complex weighted directed graphs where each edge is assigned a complex number. Necessary and sufficient conditions for the Laplacian matrix to be singular/nonsingular are derived. Our results give the relationship between the Laplacian matrix and the structure of its corresponding directed graph. Compared with the existing results, our main contribution is that our results are established without the restriction that the adjacency matrix is Hermitian. [ABSTRACT FROM AUTHOR]

For a complex n×n matrix T and a vector x ∈ Cn, we denote by σ T(x) (respectively, by r T(x)) the local spectrum (respectively, the local spectral radius) of T at x. We prove that φ: Mn → Mn linear has the property that for each T ∈ Mn there exists a nonzero xT ∈ Cn such that σ φ (T) (xT)=σ T(x T) if, and only if, there exists A ∈ Mn invertible such that either φ T) = ATA − 1 for each T ∈ Mn, or φ (T)=ATt A − 1 for each T ∈ Mn. Modulo a multiplication by a unimodular complex number, we arrive at the same conclusion by supposing that for each T ∈ Mn there exists a nonzero xT ∈Cn such that r φ (T) (xT)=rT (xT). [ABSTRACT FROM AUTHOR]

Abstract: Let be a positive semidefinite block matrix with square matrices M and N of the same order and denote . The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, If, in addition, K is Hermitian, then for any real number , while if K is skew-Hermitian, then for any real number , where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz. [Copyright &y& Elsevier]

Abstract: The energy of a digraph D is defined as , where denotes the real part of the complex number . We study in this work the energy over the set consisting of digraphs with n vertices and cycles of length . Due to the fact that the characteristic polynomial of a digraph D∈ has an expression of the form where are nonnegative integers, it is possible to define a quasi-order relation over , in such a way that the energy is increasing. Moreover, we show that the energy of a digraph decreases when an arc of a cycle of length 2 is deleted. Consequently, we obtain extremal values of the energy over sets of directed hexagonal systems, i.e. digraphs whose underlying graph is a hexagonal system. [Copyright &y& Elsevier]

Abstract: Let A be the adjacency matrix of a graph X and suppose . We view A as acting on and take the standard basis of this space to be the vectors for u in . hysicists say that we have perfect state transfer from vertex u to v at time if there is a scalar such that (Since is unitary, .) For example, if X is the d-cube and u and v are at distance d then we have perfect state transfer from u to v at time . Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from u to v if there is a complex number and, for each positive real there is a time t such that Again we necessarily have . In a recent paper Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path if and only is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for and .) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we extend the theory of pretty good state transfer. We provide what is only the second family of graphs where pretty good state transfer occurs. The graphs we use are the double-star graphs , these are trees with a vertex of degree adjacent to a vertex of degree , and all other vertices of degree one. We prove that perfect state transfer does not occur in any graph in this family. We show that if , then there is pretty good state transfer in between the two end vertices adjacent to the vertex of degree three. If , we prove that there is never perfect state transfer between the two vertices of degree at least three, and we show that there is pretty good state transfer between them if and only these vertices both have degree and is not a perfect square. Thus we find again the the existence of perfect state transfer depends on a number theoretic condition. It is also interesting that although no double stars have perfect state transfer, there are some that admit pretty good state transfer. [Copyright &y& Elsevier]

Abstract: Let be a JB∗-triple and let be a linear projection. It is proved that is an isometry for some modulus one complex number if and only if either or is hermitian. It is also proved that every rank one bicontractive projection on is hermitian. The particular case when is a C∗-algebra is discussed through several examples. [Copyright &y& Elsevier]

Abstract: The ray of a complex number is either or depending on whether is 0 or nonzero. The ray pattern of a complex matrix , denoted by , is the matrix obtained by replacing each entry of with its ray. The determinantal region of a square matrix , denoted by , is the set of the determinants of all the complex matrices with the same ray pattern as . A connected component of the set is called a determinantal regional component of . The number of determinantal regional components of is denoted by . It was proved in Shao et al. [Jia-Yu Shao, Yue Liu, Ling-Zhi Ren, The inverse problems of the determinantal regions of ray pattern and complex sign pattern matrices, Linear Algebra Appl. 416 (2006) 835–843] that for any complex square matrix . When , the two determinantal regional components are either two opposite open rays or two opposite open sectors with the angle no more than . In this paper, we prove that any square matrix with is partly decomposable if one of its determinantal regional components is an open sector with the angle less than . As a main graph theoretical technique, we also discuss a property of strongly connected digraphs. [Copyright &y& Elsevier]

Abstract: Given four complex matrices and D, where and , and given a complex number : What is the (spectral norm) distance from D to the set of matrices such that is a multiple eigenvalue of the matrixThis problem is solved when is not an eigenvalue of A. We also give a conjecture when is an eigenvalue of A. [Copyright &y& Elsevier]

Abstract: For a matrix polynomial and a given complex number , we introduce a (spectral norm) distance from to the matrix polynomials that have as an eigenvalue of geometric multiplicity at least , and a distance from to the matrix polynomials that have as a multiple eigenvalue. Then we compute the first distance and obtain bounds for the second one, constructing associated perturbations of . [Copyright &y& Elsevier]

Abstract: A companion matrix is determined by the zeros of it’s characteristic polynomial. We determine the location of the zeros which yields an elliptic numerical range. In particular we show that given any two complex numbers z 1 and z 2 there exists a third complex number z 3 such that the companion matrix of the polynomial will have elliptic numerical range with foci at z 1 and z 2. [Copyright &y& Elsevier]

Abstract: For any complex number αand any even-size skew-symmetric matrix B, we define a generalization pf α (B) of the pfaffian pf(B) which we call the α-pfaffian. The α-pfaffian is a pfaffian analogue of the α-determinant studied in [T. Shirai and Y. Takahashi, J. Funct. Anal. 205 (2003) 414–463] and [D. Vere-Jones, Linear Algebra Appl. 111 (1988) 119–124]. It gives the pfaffian at α =−1. We give some formulas for α-pfaffians and study the positivity. Further we define point processes determined by the α-pfaffian. Also we provide a linear algebraic proof of the explicit pfaffian expression obtained in [S. Matsumoto, Correlation functions of the shifted Schur measure, math.CO/0312373] for the correlation function of the shifted Schur measure. [Copyright &y& Elsevier]

We consider the group SLnF of all n×n matrices with determinant 1 over a field F. We prove that, if F is the complex number field or the real number field, every matrix A in SLnF is a product of at most two commutators of involutions. Moreover, two is the smallest such number. [Copyright &y& Elsevier]

If ζ is a nonzero complex number and P is a monic polynomial with real coefficients, let Kn(ζ;P)=(P(|r−s|)ρ|r−s|ei(r−s)φ)r,s=1n. We call the class of matrices Tn=∑jcjKn(ζj;Pj) (cj real, finite sum) generalized Kac–Murdock–Szego¨ matrices. If |ζj|<1 for all j, the family {Tn} has a generating function in C[−π,π], and Szego¨'s distribution theorem implies that the eigenvalues of Tn are distributed like the values of g as n→∞. However, Szego¨'s theorem does not apply if |ζj|&ges;1 for some j. Nevertheless, we show that in this case, provided that Pj is even if |ζj|=1, there is a function g∈C[−π,π] such that all but a finite number (independent of n) of the eigenvalues of Tn are distributed like the values of g as n→∞. We also discuss the asymptotic behavior of the remaining eigenvalues as n→∞; however, a complete resolution of this question is not yet available. [Copyright &y& Elsevier]