Fengxia Zhang

This paper studies several special least squares solutions of complex split quaternion matrix equations and . First of all, for the complex split quaternion matrix, we define its real representation by applying the semi‐tensor product (STP) and study its properties. Then, according to these properties, the MP inverse, the Kronecker product and the operator, we provide expressions for three special types of least squares solutions and their corresponding minimum norm solutions of and . Subsequently, we propose real algorithms for above minimum norm solutions and verify their efficiency by several numerical experiments. Eventually, as an application, we employ the least squares pure imaginary solution of to handle the problem of color image restoration.

By applying the real representation matrix and the Moore–Penrose inverse of the split quaternion matrix, we first study the split quaternion least squares problem with equality constraint (SQLSE) and obtain its general solution and minimum norm solution. Then, we obtain the Karush–Kuhn–Tucker equation, which is an equivalent problem of the SQLSE problem. Immediately after, we propose a numerical algorithm on the real number field for the minimum norm solution of the SQLSE problem and enumerate three examples to display its validity. Eventually, we provide an application of the SQLSE problem in the color image restoration.

This paper investigates robust set stability for the switched Boolean networks (SBNs) with arbitrary switching signal affected by one-bit function perturbation (OBFP). Firstly, the dynamics of these networks are converted into algebraic forms utilizing semi-tensor product (STP) method. Secondly, OBFP impact on the set stability of SBNs is divided into four cases. Then, by constructing a state set and defining an index vector, several necessary and sufficient conditions are provided to detect whether an SBN with arbitrary switching signal is still stable to the given set after OBFP. Finally, a biological example is proposed to demonstrate the effectiveness of the obtained theoretical results.

This paper is devoted to the condition numbers of quaternion linear system with multiple right-hand sides and the associated condition numbers of the quaternion matrix inverse as well. The explicit expressions of the unstructured and structured normwise, mixed, and componentwise condition numbers for the system are given. To reduce the computational cost of the condition numbers,compact and tight upper bounds for these condition numbers are proposed. For general sparse and badly scaled problems, numerical examples show that mixed and componentwise condition numbers are preferred than the normwise condition number for estimating the forward error of the solution, and structured condition numbers are tighter than the unstructured ones for some specific structured problems.