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...

We propose a method to compute the $R$-matrix $R$ on tensor product of Fock modules from coproduct relations in Hopf algebra. apply this quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ for which is currently not explicitly known. show that reduce single elegant equation $R$. Using theory symmetric Macdonald polynomials we provides recursive formula matrix elements

We consider a strongly regular graph, G, with adjacency matrix A, and associate a three dimensional Euclidean Jordan algebra to A. Then, by considering convergent series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of the Euclidean Jordan algebra associated to A, we establish new admissibility conditions for the existence of strongly regular grap...

Let B be a block of an Iwahori–Hecke algebra or q-Schur algebra of the symmetric group. The decomposition matrix for B may be obtained from the decomposition matrix of the corresponding block B′ in infinite characteristic by post-multiplying by an adjustment matrix; since (by a deep theorem of Ariki) there is an algorithm for computing the decomposition matrix for B′, the hard part of the decom...

Modern system dynamics is based upon a matrix representation of the dynamic equations governing the system behavior. A basic understanding of elementary matrix algebra is essential for the analysis of state-space formulated systems. A full discussion of linear algebra is beyond the scope of this note and only a brief summary is presented here. The reader is referred to a text on linear algebra,...

We consider (max,+)-algebra products of random matrices, which arise from performance evaluation of acyclic fork-join queueing networks. A new algebraic technique to examine properties of the product and investigate its limiting behaviour is proposed based on an extension of the standard matrix (max,+)-algebra by endowing it with the ordinary matrix addition as an external operation. As an appl...

Transitivity of generalized fuzzy matrices over a special type of semiring is considered. The semiring is called incline algebra which generalizes Boolean algebra, fuzzy algebra, and distributive lattice. This paper studies the transitive incline matrices in detail. The transitive closure of an incline matrix is studied, and the convergence for powers of transitive incline matrices is considere...

A differential central simple algebra, and in particular a differential matrix algebra, over a differential field K with constants C can be trivialized by a Picard–Vessiot (differential Galois) extension E. In the matrix algebra case, there is a correspondence between K algebras trivialized by E and representations of the Galois group of E over K in PGLn(C), which can be interpreted as cocyles ...

It is known that a family of transfer matrix functional equations, the T-system, can be compactly written in terms of the Cartan matrix of a simple Lie algebra. We formally replace this Cartan matrix of a simple Lie algebra with that of an affine Lie algebra, and then we obtain a system of functional equations different from the T-system. It may be viewed as an X (a) n type affine Toda field eq...

Superimposed D–branes have matrix–valued functions as their transverse coordinates, since the latter take values in the Lie algebra of the gauge group inside the stack of coincident branes. This leads to considering a classical dynamics where the multiplication law for coordinates and/or momenta, being given by matrix multiplication, is nonabelian. Quantisation further introduces noncommutativi...