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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]

In this paper, two new algorithms for computing the Weighted Moore–Penrose inverse A M , N † of a general matrix A for weights M and N which are based on elementary row and column operations on two appropriate block partitioned matrices are introduced and investigated. The computational complexity of the introduced two algorithms is analyzed in detail. These two algorithms proposed in this paper are always faster than those in Sheng and Chen (2013) and Ji (2014), respectively, by comparing their computational complexities. In the end, an example is presented to demonstrate the two new algorithms. [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]

Abstract In this paper, we present functional definitions of all types of generalized inverses related to the {1}-inverse, which is a continuation of the work of Campbell and Meyer (2009). According to these functional definitions, we further derive novel representations for all types of generalized inverses related to the {1}-inverse in terms of the bases for R (A *), N (A) and N (A *). Based on these representations, we present the corresponding algorithm for computing various generalized inverses related to the {1}-inverse of a matrix and analyze the computational complexity of our algorithm for a constant matrix. Finally, we implement our algorithm and several known algorithms for symbolic computation of the Moore-–Penrose inverse in the symbolic computational package MATHEMATICA and compare their running times. Numerical experiments show that our algorithm outperforms these known algorithms when applied to compute the Moore–Penrose inverse of one-variable rational matrices, but is not the best choice for two-variable rational matrices in practice. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the theory on direct and inverse spectral problems for Jacobi matrices is revisited in a kind of pseudo-Jacobi matrices J (n , r , β) with a mixed path as its graph in the non-self-adjoint setting. In this context, a sign change in one of the nondiagonal entries of the matrix yields strong perturbations in its spectral properties. The reconstruction of a pseudo-Jacobi matrix from its spectrum and the spectra of two complementary principal matrices is investigated. An algorithm for the reconstruction of matrices from prescribed spectral data is provided and illustrative numerical experiments are performed. [ABSTRACT FROM AUTHOR]

Highlights • Two new Taylor algorithms for the computation of the matrix exponential are proposed. • They are based on matrix polynomial evaluation methods more efficient than Paterson–Stockmeyer method. • In tests they showed higher efficiency and accuracy than state-of-the-art Padé based methods. • They also showed a higher efficiency than state-of-the-art Taylor methods, and higher accuracy in some tests. Abstract This paper presents new Taylor algorithms for the computation of the matrix exponential based on recent new matrix polynomial evaluation methods. Those methods are more efficient than the well known Paterson–Stockmeyer method. The cost of the proposed algorithms is reduced with respect to previous algorithms based on Taylor approximations. Tests have been performed to compare the MATLAB implementations of the new algorithms to a state-of-the-art Padé algorithm for the computation of the matrix exponential, providing higher accuracy and cost performances. [ABSTRACT FROM AUTHOR]

The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Finding efficient algorithms for computing characteristic polynomial of graphs is an active area of research and for some graph classes, like threshold graphs, there exist very fast algorithms which exploit combinatorial structure of the graphs. In this paper, we put forward some novel ideas based on divisor technique to obtain fast algorithms for computing the characteristic polynomial of threshold and chain graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the static output feedback set stabilization for context-sensitive probabilistic Boolean control networks (CS-PBCNs) via the semi-tensor product of matrices. An algorithm for finding the largest control invariant set with probability one is obtained by the algebraic representations of logical dynamics. Based on the analysis of the set stabilization, necessary and sufficient conditions for S -stabilization are obtained. Static output feedback controllers are designed to achieve S -stabilization for a CS-PBCN. At last, examples to study metastatic melanoma are given to show the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations A 1 X = C 1 , Y B 1 = D 1 , A 2 W = C 2 , Z B 2 = D 2 , A 3 V = C 3 , V B 3 = C 4 , A 4 V B 4 = C 5 , A 5 X + Y B 5 + C 6 W + Z D 6 + E 6 V F 6 = G 6 , and constitute an expression of the general solution to the system when it is solvable. The outcomes of this paper encompass some recognized results in the collected works. In addition, we establish an algorithm and a numerical example to illustrate the theory constructed in the paper. [ABSTRACT FROM AUTHOR]

First of all, by characterizing solutions of the quaternion equation a x = x ˜ b , this paper studies consimilarity of quaternions and some related consequences. For the important role of coneigenvalues in consimilarity tranformations of quaternion matrices, this paper further derives the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of the corresponding real representation matrix. Then, based on the real representation matrix, an effective algorithm is presented to calculate all coneigenvalues and the associated coneigenvectors of a quaternion matrix. Finally, two numerical examples are given to verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new numerical algorithm for solving the Riesz tempered space fractional diffusion equation. The stability and convergence of the numerical scheme are discussed via the technique of matrix analysis. Finally, numerical experiments are performed to confirm the effectiveness of our numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fuzzy SV- k -modes algorithm that uses the fuzzy k -modes clustering process to cluster categorical data with set-valued attributes. In the proposed algorithm, we use Jaccard coefficient to measure the dissimilarity between two objects and represent the center of a cluster with set-valued modes. A heuristic update way of cluster prototype is developed for the fuzzy partition matrix. These extensions make the fuzzy SV- k -modes algorithm can cluster categorical data with single-valued and set-valued attributes together and the fuzzy k -modes algorithm is its special case. Experimental results on the synthetic data sets and the three real data sets from different applications have shown the efficiency and effectiveness of the fuzzy SV- k -modes algorithm. [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]

In this paper, we investigate the nonlinear matrix equations X + A * X q A = Q ( 0 < q ≤ 1 ) and X − A * X q A = Q ( q > 1 ) . Necessary and sufficient conditions for the (unique) existence of Hermitian positive definite solutions of these equations are derived. Effective iterative algorithms are also provided to obtain the unique solution of X + A * X q A = Q ( 0 < q ≤ 1 ) and the minimal and maximal Hermitian positive definite solutions of X − A * X q A = Q ( q > 1 ) . Numerical examples are included to illustrate the efficiency of the presented iteration algorithms. [ABSTRACT FROM AUTHOR]

In this paper, by applying particular structure of the real representations of quaternion matrices and the Moore–Penrose generalized inverse, we derive the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X = B . The resulting formulas only involve real matrices, which are simpler than those reported in (Yuan et al., 2013). The corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices, therefore are very efficient and easily understood. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a fast proximal point algorithm (PPA) is proposed for solving ℓ 1 -minimization problem arising from compressed sensing. The proposed algorithm can be regarded as a new adaptive version of customized proximal point algorithm, which is based on a novel decomposition for the given nonsymmetric proximal matrix M . Since the proposed method is also a special case of the PPA-based contraction method, its global convergence can be established using the framework of a contraction method. Numerical results illustrate that the proposed algorithm outperforms some existing proximal point algorithms for sparse signal reconstruction. [ABSTRACT FROM AUTHOR]

In this paper we consider relations between the principal iterations from the Padé family of Kenney and Laub for computing the matrix sign function and the principal iterations from the reciprocal Padé family of Greco, Iannazzo and Poloni, and the dual Padé family of Ziȩtak. We show global convergence of the principal reciprocal Padé iterations and the principal dual Padé iterations. We adopt the dual Padé family of iterations, to which the Newton method belongs, for computing the unitary polar factor of a nonsingular matrix. We present numerical experiments with the scaled Newton method of Higham for computing the unitary polar factor, which show how the quality of the computed inverses of matrices in the scaled Newton method affects the accuracy of the computed polar factorization of a nonsingular matrix. It indicates that assumptions, under which the backward stability of the scaled Newton method has been proved in the literature, cannot be weaker. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new algorithm for the computation of the common solution to the complex conjugate matrix equation system A i X B i + C i X ¯ D i = E i , i = 1 , 2 , … , N . The algorithms only need finite iteration steps, requiring only real computation. Furthermore, the algorithm can be extended to solve a more general complex matrix equation system. Two numerical examples are given to illustrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The sparse representation of a multiple measurement vector (MMV) is an important problem in compressed sensing theory, the old alternating direction method (ADM) is an optimization algorithm that has recently become very popular due to its capabilities to solve large-scale or distributed problems. The MMV–ADM algorithm to solve the MMV problem by ADM has been proposed by H. Lu, et al. (2011)[24], but the theoretical result about the convergence of matrix iteration sequence generated by the algorithm is left as a future research topic. In this paper, based on the subdifferential property of the two-norm for vector, a shrink operator associated with matrix is established. By using the operator, a convergence theorem is proved, which shows the MMV–ADM algorithm can recover the jointly sparse vectors. [ABSTRACT FROM AUTHOR]

This paper focuses on the group consensus issue of multi-agent systems, where the agents in a network can reach more than one consistent values asymptotically. A rotation matrix is introduced to an existing consensus algorithm for single-integrator dynamics. Based on algebraic matrix theories, graph theories and the properties of Kronecker product, some necessary and sufficient criteria for the group consensus are derived, where we show that both the eigenvalue distribution of the Laplacian matrix and the Euler angle of the rotation matrix play an important role in achieving group consensus. Simulated results are presented to demonstrate the theoretical 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]

• Derivation of rank-Greville algorithm, maintaining a general rank factorization. • Exploiting rank-deficiency to reach an asymptotic computational complexity of O(mr) for updating a least-squares solution when adding an observation. • Publically available implementation in Python3, using Numpy. Updating a linear least-squares solution can be critical for near real-time signal-processing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A ∈ R n × m with rank r. In this paper, we explicitly derive a similar formula by maintaining a general rank factorization, which we call rank-Greville. Based on this formula, we implemented a recursive least-squares algorithm exploiting the rank-deficiency of A , achieving the update of the minimum-norm least-squares solution in O (m r) operations and, therefore, solving the linear least-squares problem from scratch in O (n m r) operations. We empirically confirmed that this algorithm displays a better asymptotic time complexity than LAPACK solvers for rank-deficient matrices. The numerical stability of rank-Greville was found to be comparable to Cholesky-based solvers. Nonetheless, our implementation supports exact numerical representations of rationals, due to its remarkable algebraic simplicity. [ABSTRACT FROM AUTHOR]

The spark is an important parameter to evaluate the recovery performance of measurement matrices in compressed sensing. This paper presents an effective algorithm to calculate the upper bound of the spark of sparse binary measurement matrices. Particularly, the spark of some binary measurement matrices can be accurately calculated by using our algorithm. [ABSTRACT FROM AUTHOR]