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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]

• 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]

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 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]