The Discrete Lyapunov Equation of The Orthogonal Matrix in Semiring

Abstract

Semiring is an algebraic structure of $(S, +, \times)$. Similar to a ring, but without the condition that each element must have an inverse to the adding operation. The forms $(S, +)$ and $(S, \times)$ are semigroups that satisfy the distributive law of multiplication and addition. In matrix theory, there is a term known as the Kronecker product. This operation transforms two matrices into a larger matrix containing all possible products of the entries in the two matrices. This Kronecker product has several properties often used to solve the complex problems of linear algebra and its applications. The Kronecker product is related to the Lyapunov equation of a linear system. Based on previous research in the Lyapunov equation in conventional linear algebra, this paper will describe the characteristics of the Lyapunov equation in a semiring linear system in terms of the Kronecker product.