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This paper is concerned with the solution to the least squares problem for the quaternion Sylvester tensor equation. An iterative algorithm based on tensor format is presented to solve this problem. The convergence properties of the proposed iterative method are studied. We also consider the best approximate problem related to quaternion Sylvester tensor equation. Numerical examples are provided to confirm the theoretical results, which demonstrate that the proposed algorithm is effective and feasible for solving the least squares problem of the quaternion Sylvester tensor equation. [ABSTRACT FROM AUTHOR]

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra. [ABSTRACT FROM PUBLISHER]

In this paper, we discuss Hermitian solutions of split quaternion matrix equation $$AXB+CXD=E,$$ where X is an unknown split quaternion Hermitian matrix, and A, B, C, D, E are known split quaternion matrices with suitable size. The objective of this paper is to establish a necessary and sufficient condition for the existence of a solution and a solution formulas. Moreover, we provide numerical algorithms and numerical examples to exemplify the results. [ABSTRACT FROM AUTHOR]

Let R and H m × n stand, respectively, for the real number field and the set of all m × n matrices over the real quaternion algebra H = { a 0 + a 1 i + a 2 j + a 3 k | i 2 = j 2 = k 2 = ijk = − 1 , a 0 , a 1 , a 2 , a 3 ∈ R } . For η ∈ { i, j, k }, a real quaternion matrix A ∈ H n × n is said to be η -Hermitian if A η * = A where A η * = − η A * η , and A * stands for the conjugate transpose of A , arising in widely linear modeling. We present a simultaneous decomposition for a set of nine real quaternion matrices involving η -Hermicity with compatible sizes: A i ∈ H p i × t i , B i ∈ H p i × t i + 1 , and C i ∈ H p i × p i , where C i are η -Hermitian matrices, ( i = 1 , 2 , 3 ) . As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an η -Hermitian solution to the system of coupled real quaternion matrix equations A i X i A i η * + B i X i + 1 B i η * = C i , ( i = 1 , 2 , 3 ) , and provide an expression of the general η -Hermitian solutions to this system. Moreover, we derive the rank bounds of the general η -Hermitian solutions to the above-mentioned system using ranks of the given matrices A i , B i , and C i as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider an expression of the general solution to the classical system of matrix equationsWe present a necessary and sufficient condition for the existence of a solution to the system by using generalized inverses. We give an expression of the general solution to the system when it is solvable. As applications, we derive some necessary and sufficient conditions for the consistence to the systemand the systemwheremeans conjugate transpose. We also give the expressions of the general solutions to the systems. [ABSTRACT FROM AUTHOR]

In this paper, we give an algorithm to find the roots of the octonionic quadratic equationx2 + bx + c = 0, and develop a Matlab package to find solutions. We also discuss how to find the roots of some other octonion quadratic equations, such as an algorithm is given for finding the roots of the octonion quadratic equationxax + bx + c = 0. [ABSTRACT FROM AUTHOR]

For anyA=A1+A2j∈Qn×nand η∈i, j, k, denoteAη H=−ηAHη. IfAη H=A, Ais called an $\eta$-Hermitian matrix. IfAη H=−A, Ais called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices byηHQn×nandηAQn×n, respectively. By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equationAXB+CYD=EoverX∈ηHQn×nandY∈ηAQn×n. [ABSTRACT FROM PUBLISHER]