This paper is an updated and extended version of the paper “The QR decomposition and the singular value decomposition in the symmetrized max-plus algebra” (by B. De Schutter and B. De Moor, SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 2, pp. 378–406, April 1998). The max-plus algebra, which has maximization and addition as its basic operations, can be used to describe and anal...

In this paper circulant matrices in max-plus algebra are presented. Circulant matrices are special form of matrices which are entered by vector of inputs. For special types of matrices such as circulant matrices, the computation can often be performed in the simpler way than in the general case. The so-called max-plus algebra is useful for investigation of discrete events systems and the sequen...

The max-plus algebra has maximization and addition as basic operations, and can be used to model a certain class of discrete event systems. In contrast to linear algebra and linear system theory many fundamental problems in the max-plus algebra and in max-plus-algebraic system theory still need to be solved. In this paper we discuss max-plus-algebraic analogues of some basic matrix decompositio...

The max-plus semiring Rmax is the set R∪{−∞}, equipped with the addition (a, b) 7→ max(a, b) and the multiplication (a, b) 7→ a + b. The identity element for the addition, zero, is −∞, and the identity element for the multiplication, unit, is 0. To illuminate the linear algebraic nature of the results, the generic notations +, , × (or concatenation), 0 and 1 are used for the addition, the sum, ...

Periods of matrix power sequences in max-drast fuzzy algebra and methods of their computation are considered. Matrix power sequences occur in the theory of complex fuzzy systems with transition matrix in max-t algebra, where t is a given triangular fuzzy norm. Interpretation of a complex system in max-drast algebra reflects the situation when extreme demands are put on the reliability of the sy...

First we establish a connection between the field of the real numbers and the extended max algebra, based on asymptotic equivalences. Next we propose a further extension of the extended max algebra that will correspond to the field of the complex numbers. Finally we use the analogy between the field of the real numbers and the extended max algebra to define the singular value decomposition of a...

Let a ⊕ b = max(a, b), a ⊗ b = a + b for a, b ∈ R := R ∪ {−∞}. By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machinescheduling, information technology and ...

Max-times algebra, sometimes known as subtropical algebra, is a semi-ring over the nonnegative real numbers where the addition operation is the max function and the multiplication is the standard one. Factorizing a nonnegative matrix over the maxtimes algebra, instead of the standard (nonnegative) one, allows us to find structures and regularities that cannot be easily expressed in the standard...

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we ...

Let a ⊕ b = max(a, b), a ⊗ b = a + b for a, b ∈ R := R ∪ {−∞}. By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and...