ON THE MATRIX A⊗A* OF SIZE 2x2 IN MAX-PLUS ALGEBRA

Abstract

Max-plus algebra is an analogue of linear algebra, developed for a pair of operations denoted as ⊕ and
⊗, which we call max-algebraic addition and multiplication, respectively. These operations, when extended to
vectors and matrices, redefine the classical approach to linear algebraic operations. Specifically, replacing addition
and multiplication with max-plus operations results in a novel mathematical framework that enables nonlinear
problems to be analyzed and solved in a linear manner. We often call max-plus algebra simply max-algebra (we can
also talk about min-plus algebra, or max-time algebra, but those structures are not the subject of this paper). This
mathematical area began its development in the 1960s, and since then, a substantial body of research has been
published. Many of these papers address topics central to classical linear algebra, such as eigenvalues and
eigenvectors, solving systems of equations, matrix ranks, and matrix powers. Max-algebra has also found practical
applications in fields like manufacturing, transportation, traffic light control, resource allocation, and the natural
sciences. However, one notable limitation in max-algebra is the nonexistence of inverse matrices in the general case
- they are only defined for generalized permutation matrices. This complicates problem-solving, as inverse matrices
are indispensable in classical linear algebra for tasks like solving systems of equations. Consequently, new tools and
methodologies must be developed to address foundational issues within max-algebra. One such tool, beside dual
operations, max-algebraic permanent, and maximum cycle mean, is the conjugate matrix, defined as the negative
transpose matrix of a given matrix. The conjugate matrix has appeared in pioneering works on max-algebra. Also,
matrices derived from the max-algebraic sum or product of a matrix and its conjugate exhibit unique properties. For
instance, a matrix resulting from the max-algebraic product of a given matrix and its conjugate matrix is a square
matrix with a zero diagonal. With this in mind, this paper examines the following question: given a square matrix
with a zero diagonal, is it possible to determine a matrix ???? such that the max-algebraic product of ???? and its
conjugate matrix equals the given matrix? Due to the complexity of this problem, we focus on the simplest case:
when a square matrix of dimensions 2 × 2 is given. This serves as a basis for further exploration and the derivation
of new insights regarding matrices of higher dimensions. A square matrix of dimensions 2 × 2 with a zero diagonal
inherently contains two off-diagonal elements. Consequently, as we shall demonstrate, solving the problem requires
examining two distinct cases. These cases are explained through theorem along with a proof. An example
illustrating the proven facts is also given in the paper.