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The sole purpose of this short note is to correct false claims made in the paper "A new approach for finding the determinant of matrices" by Omid Rezaifar and Hossein Rezaee (Appl. Math. Comput. 188 (2007) 1445–1454). [ABSTRACT FROM AUTHOR]

An oriented circulant graph is called integral if all eigenvalues of its Hermitian adjacency matrix are integers. The main purpose of this paper is to investigate the existence of perfect state transfer (PST for short) and multiple state transfer (MST for short) on integral oriented circulant graphs. Specifically, a characterization of PST and MST on integral oriented circulant graphs is provided. As an application, we also obtain a closed-form expression for the number of integral oriented circulant graphs having PST and MST. • Use Ramanujan's sum to characterize the eigenvalues of integral oriented circulant graphs and give the complete formula. • The paper provides a characterization of PST and MST on integral oriented circulant graphs. • Closed-form expression for the number of graphs with PST and MST on integral oriented circulant graphs. [ABSTRACT FROM AUTHOR]

In this paper, by means of sliding mode control (SMC), the problem resulted by local input matrices is investigated for fuzzy markovian jumping system. The stability of the considered fuzzy Markovian system with multiple sub-input matrices can be fulfilled using the method in this paper. First, the state equation of the considered system is transformed according to the input matrices. Second, according to the state equation, sliding mode surface is constructed. This surface contains several sub-surfaces. It can deal with this kind of fuzzy Markovian system with multiple sub-input matrices and uncertainties. And another characteristic is that there is not the process of reaching the sliding mode surface; It can settle the problem resulted by Markovian jumping and sliding mode method together. Third, by use of the Cramer's rule, a criterion is provided to judge the existence of sliding mode dynamics equation. A controller containing several sub-controllers components is designed. And these sub-controllers should keep the considered system on the several sub-surfaces and not leave it. At last, simulations are provided to illustrate the validity of the method in this paper. [ABSTRACT FROM AUTHOR]

In recent years, the reduced biquaternion algebras have been widely used in color image processing problems and in the field of electromagnetism. This paper studies eigen-problems of reduced biquaternion matrices by means of a complex representation of a reduced biquaternion matrix and derives new algebraic algorithms to find the eigenvalues and eigenvectors of reduced biquaternion matrices. This paper also concludes that the number of eigenvalues of an n × n reduced biquaternion matrix is infinite. In addition, the proposed algebraic algorithms are shown to be effective in application to a color face recognition problem. • The eigen-problems of reduced biquaternion matrices are further studied based on the complex representation form. • Propose new algebraic algorithms for finding the eigenvalues and the eigenvectors of a reduced biquaternion matrix. • An n × n reduced biquaternion matrix has infinite eigenvalues. • There are multiple eigenvalues corresponding to the same eigenvector of a reduced biquaternion matrix. • The proposed method is more comprehensive and can find more eigenvalues of a reduced biquaternion matrix. [ABSTRACT FROM AUTHOR]

An element a in a Banach algebra A has g-Drazin inverse if there exists b ∈ A such that a b = b a , b = b a b and a − a 2 b ∈ A q n i l. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix ( E I F 0 ). We thereby solve a wider kind of singular differential equations posed by Campbell (1983) [2]. [ABSTRACT FROM AUTHOR]

• Algebra matrix method integrates the structural factor and the state information of social networks. • Monte Carlo simulation experiments verify the effectiveness of the algebra matrix evaluation. • The evaluation deviations of diffusion threshold are shown best performance comparing with three popular mean field methods. • The weighted average strategy is proposed as applications. Tacit knowledge is the knowledge existing in human brain which is not easy to be recorded or quantified, and often is learned in the face-to-face interactions. The tacit knowledge diffusion depends on the decision-making of tacit knowledge owners, and the expression of explicit knowledge carriers. However, the comprehensive influence of the tacit knowledge owners, explicit knowledge carriers and the relations of them were not attracted enough attention. In this paper, an algebra matrix method is used to integrate the multidimensional information of network structures and the nodes' states. By the algebra matrix method, the diffusion threshold of the tacit knowledge is calculated, which is called algebra matrix evaluation. This evaluation method is proven to be effective by comparing with Monte Carlo simulations on three types of artificial networks and five reals. With applications of the algebra matrix evaluation, we construct a co-author network according the data of the academic papers from 1980 to 2017 on Aminer platform, and define states of tacit knowledge owners and the explicit knowledge carriers by the scholar's career lengths and the paper's cited quantities respectively. It is found that the thresholds of tacit knowledge diffusion are decreasing with the expansions of the scale of the largest connected components, whether tacit knowledge diffuses in the co-author networks or in the largest connected components. And with the evolution of cumulative co-author network, the diffusion thresholds of tacit knowledge in the largest connected component decrease in ladder-like with unequal steps. Furthermore, it is find ignoring the state factor will lead to the deviation in the evaluation of tacit knowledge diffusion thresholds, which is 16.33% in the largest connected components and 45.07% in the whole network. [ABSTRACT FROM AUTHOR]

• In this paper, we first propose the extensional eigenvalues of graphs, which generalizes almost all other types of eigenvalues of graphs. • We present some basic properties of extensional eigenvalues, which also hold for classical eigenvalues. • Our method provides a consistent method that is valid for all types of graph matrices. • This paper may open a new door in spectral graph theory. Assume that G is a graph on n vertices with associated symmetric matrix M and K a positive definite symmetric matrix of order n. If there exists 0 ≠ x ∈ R n such that M x = λ K x , then λ is called an extensional eigenvalue of G with respect to K. This concept generalizes some classic graph eigenvalue problems of certain matrices such as the adjacency matrix, the Laplacian matrix, the diffusion matrix, and so on. In this paper, we study the extensional eigenvalues of graphs. We develop some basic theories about extensional eigenvalues and present some connections between extensional eigenvalues and the structure of graphs. [ABSTRACT FROM AUTHOR]

This paper connects two attractive topics in applied mathematics, r -circulant matrices and the Chebyshev polynomials. The r -circulant matrices whose entries are the Chebyshev polynomials of the first or second kind are considered. Then, estimates for spectral norm bounds of such matrices are presented. The relevance of the obtained results was verified by applying them to some of the previous results on r -circulant matrices involving various integer sequences. The acquired results justify the usefulness of the applied approach. [ABSTRACT FROM AUTHOR]

A structured eigen-problem A x + F (x) = λ x is studied in this paper, where in applications A ∈ R n × n is an irreducible Stieltjes matrix. Under certain restrictions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of A , the Newton-like algorithm for this eigen-problem is well defined and converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method. [ABSTRACT FROM AUTHOR]

• Derivation of several original representations for the Moore–Penrose inverse of particular columnwise partitioned matrices, with emphasis on those matrices which are of applicable value. • Introduction of an algorithm to calculate the Moore–Penrose inverse of columnwise partitioned matrices, along with an example demonstrating its execution. • Formulation of a number of original facts concerned with the Moore–Penrose inverse of columnwise partitioned matrices. The paper revisits the considerations carried out in [J.K. Baksalary, O.M. Baksalary, Linear Algebra Appl. 421 (2007) 16–23], where particular formulae for the Moore–Penrose inverse of a columnwise partitioned matrix were derived. An impuls to reconsider these investigations originated from a number of recently published articles in which the results established by Baksalary and Baksalary were utilized in different research areas of applicable background. In the present paper several not exposed so far consequences of the results derived in the recalled paper are unveiled, with an emphasis placed on revealing their underlying applicability capabilities. A special attention is paid to the computational aspects of the Moore–Penrose inverse determination. [ABSTRACT FROM AUTHOR]

• This article presents a new iterative family having ninth-order convergence for computing the generalized outer inverse. • The proposed scheme requires only seven matrix multiplications at each iterations, but for the specific parameters, it uses only five matrix multi- plications. • The detailed theoretical convergence analysis is presented. • A wide range of random numerical examples is included for comparing the proposed iterations with existing methods. • It is demonstrated that the proposed methods are computationally effcient and provides an attractive alternative for computing the generalized inverse in practice. This paper presents a new iterative family for computing the generalized outer inverse with a prescribed range and null space of a given complex matrix. It is proved that the proposed methods achieve at least ninth-order of convergence. In general, the improved formulation of scheme uses only seven matrix multiplications at each iteration, but for the specific parameters, it uses only five matrix multiplications. The theoretical discussion on computational efficiency index is presented. Further, numerical results obtained are compared with existing robust methods to verify the theoretical analysis and higher computational efficiency. Several numerical examples, including rectangular, rank-deficient, large sparse, and non-singular matrices from the matrix computation tool-box (mctoolbox) are included. Further, the performance of methods is measured on randomly generated singular matrices and boundary value problem. It is demonstrated that the presented scheme gives improved results than the existing Schulz-type iterative methods for calculating the generalized inverse. [ABSTRACT FROM AUTHOR]

• The distribution of eigenvalues for Seidel matrices of threshold graphs by its building string is investigated. • The inertia of the Seidel matrix of a threshold graph can be represented by its building string. • We show that the interval (− 2 , 2) contains no eigenvalue of Seidel matrix of a threshold graphs, except for − 1 and 1. The distribution of eigenvalues for Seidel matrices of threshold graphs is investigated in this paper. We show that there is no eigenvalue of Seidel matrices of threshold graphs in the interval (− 2 , 2) except for − 1 and 1. We also determine the inertia of the Seidel matrix of a threshold graph in terms of its binary string. [ABSTRACT FROM AUTHOR]

• The spectral properties of a newer graph matrix, the anti-adjacency matrix (or eccentricity matrix), are investigated. • New families of graphs determined by their anti-adjacency spectra are discovered. • The Vita Theorem is found to be very useful in the proofs of main results. The anti-adjacency matrix (or eccentricity matrix) of a graph is obtained from its distance matrix by retaining for each row and each column only the largest distances. This matrix can be viewed as the opposite of the adjacency matrix, which is, on the contrary, obtained from the distance matrix of a graph by keeping for each row and each column only the distances being 1. In this paper, we prove that the graphs with exactly one positive anti-adjacency eigenvalue are determined by the anti-adjacency spectra. As corollaries, the well-known (generalized) friendship graphs and windmill graphs are shown to be determined by their anti-adjacency spectra. [ABSTRACT FROM AUTHOR]

• A construction of (n , n , n − 1 , n − 1 , n − 2) -BIBD based on finite set. The incidence matrix H of (n , n , n − 1 , n − 1 , n − 2)-BIBD based on finite set. • A method of combining vertical expansions and horizontal expansions to construct measurement matrices. • An approach for the embedding matrix with low coherence. In signal processing, compressed sensing (CS) can be used to acquire and reconstruct sparse signals. This paper presents a method of combining vertical expansions and horizontal expansions to construct measurement matrices. Firstly, we give a construction of (n , n , n − 1 , n − 1 , n − 2) -BIBD based on finite set. It is important to estimate recovery performance of measurement matrices in terms of coherence, and it is found that the incidence matrix H of (n , n , n − 1 , n − 1 , n − 2) -BIBD is not suitable as a measurement matrix in CS. We present an optimal method of combining vertical expansions and horizontal expansions for addressing this problem. These two extensions provide a new perspective for the construction of measurement matrices. Vertical expansions ensure that the matrix has low coherence. Horizontal expansions ensure that the matrix is more suitable as a measurement matrix in CS because of sizes and coherence. Finally, compared with several typical matrices, our matrices have better recovery performance under OMP and IST by the simulation experiments. [ABSTRACT FROM AUTHOR]

• The column unitary solutions to the matrix equation A X B = D and the system of matrix equations A X = C , X B = D are considered. • The solvability conditions for the existence and the explicit expressions for the general solution are obtained. • Simple recipes for the numerical computations are provided. In this paper, the solvability conditions for the matrix equation A X B = D and a pair of matrix equations A X = C , X B = D with the constraint X * X = I p are deduced by applying the spectral and singular value decompositions of matrices, and the expressions of the general solutions to these matrix equations are also provided. Furthermore, the associated optimal approximate problems to the given matrices are discussed and the optimal approximate solutions are derived. Finally, two numerical experiments are given to validate the accuracy of the results. [ABSTRACT FROM AUTHOR]

• In this paper, we proposed an algorithm to solve mixed solutions of the matrix Equation ∑ i = 1 t A i X i B i = E with sub-matrix constraints. We also prove that the iterative solution sequence generated by the algorithm is convergent. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case, for both small-scale and large-scale matrices. The work belongs to the field of numerical algebra, and has been widely concerned. We put forward and analyze in details an iterative method to find the mixed solutions of a matrix equation with sub-matrix constraints. The convergence of the approximated solution sequence generated by the iterative method is investigated, showing that if the constrained matrix equation is consistent, the mixed solution group can be obtained after a finite number of iterations. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case for both small-scale and large-scale matrices. [ABSTRACT FROM AUTHOR]

• We illustrate a recent method for proving identities of matrices and linear operators. • Computations are done in an abstract ring ignoring domains and codomains. • The theory ensures that identities are also valid in terms of matrices or operators. • We prove and generalize Hartwig's result on three Moore-Penrose inverses. • Both hand-made and computer-assisted proofs with the package OperatorGB are discussed. When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig's well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig's result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials. [ABSTRACT FROM AUTHOR]

• Based on the iteration frameworks [6], by introducing a tunable parameter ω , several relaxed iteration methods are proposed for solving the matrix equation AXB = C, and their convergence properties are analyzed in detail. • Combining the iteration methods in [6] with the tunable parameter ω , some relaxed preconditioned iteration methods are also given. • The choices of the parameter ω in the relaxed iteration methods are discussed, and the optimal choices of the parameter ω to achieve the fastest convergence rate are also obtained for some special cases. • Based on the matrix splittings B T = D 1 − L 1 − U 1 , A = D 2 − L 2 − U 2 ,three relaxed iteration methods are constructed, which are the relaxed Jacobi-type iteration method, relaxed SSOR-type iteration method and relaxed bilateral Jacobi-type iteration method, respectively. Moreover, the optimal parameter ω can be calculated in these iteration methods. • For the general matrix equation AXB=C, we consider how to solve it by using the proposed relaxed iteration methods, and investigate how to reduce their computational costs based on the Hessenberg decompositions of the corresponding matrices. • Numerical examples show that our proposed algorithms are more efficient than GI method [4] HSS method [17] and the iteration methods in [6], respectively. In this paper, based on the iteration frameworks [6], several relaxed iteration methods are proposed for solving the matrix equation A X B = C by introducing a tunable parameter ω , and their convergence properties are analyzed in detail. Moreover, the optimal choices of the parameter ω to achieve the fastest convergence rate are also obtained for some special cases. Finally, numerical experiments are carried out to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices, resulting in the total time complexity O (n 4) , where n is the order of the matrix. We present a method for general interval matrices as well as its modification for symmetric interval matrices. In the second part of the paper, we apply the spectral decomposition to computing powers of interval matrices, which is our second goal. Numerical results suggest that a simple binary exponentiation is more efficient for smaller exponents, but our approach becomes better when computing higher powers or powers of a special type of matrices. In particular, we consider symmetric interval and circulant interval matrices. In both cases we utilize some properties of the corresponding classes of matrices to make the power computation more efficient. [ABSTRACT FROM AUTHOR]

Networks in which connections change over time arise in many applications, e.g., when modeling phone calls and flights between airports. This paper discusses new ways to define adjacency matrices associated with this kind of networks. We propose that dynamic networks be modeled with the aid of block upper triangular adjacency matrices. Both modeling and computational aspects are discussed. Several applications to real dynamic networks are presented and illustrate the advantages of the proposed method when compared with an available approach. [ABSTRACT FROM AUTHOR]