Related papers: Some thoughts about matrix coordinate transformati…
We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.
Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this paper aims to answer the following question: Can a transformation of the generalized coordinates under which the actuators…
A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed in q-alg/9609011 In this paper we give an outline of the construction of a noncommutative analogy of the algebra of partial…
The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a…
The vector transform operators are investigated; these operators are used at the solution of boundary value problems in piecewise homogeneous spherically symmetric areas. In particular, examples of transformation operators for vector…
This article represents a major step in the unification of the theory of algebraic, topological and singular transition matrices by introducing a definition which is a generalization that encompasses all of the previous three. When this…
We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are…
Transformer-based models excel in various tasks but their generalization capabilities, especially in arithmetic reasoning, remain incompletely understood. Arithmetic tasks provide a controlled framework to explore these capabilities, yet…
We derive and discuss a technique for manipulating power series which is complementary to standard procedures. We begin with the translation operator, but we express the operator as an infinite product instead of expanding it as a series…
We study a filtered generalization of the operation of elementary modification of vector bundles. The generalization is motivated by applications to the degeneration theory of linear systems.
This paper investigates the failure cases and out-of-distribution behavior of transformers trained on matrix inversion and eigenvalue decomposition. I show that incorrect model predictions still retain deep mathematical properties of the…
A recent body of work has demonstrated that Transformer embeddings can be linearly decomposed into well-defined sums of factors, that can in turn be related to specific network inputs or components. There is however still a dearth of work…
We show that the change of basis matrices of a set of $m$ bases of a finite vector space is a connected groupoid of order $m^2$. We define a general method to express the elements of change of basis matrices as algebraic expressions using…
We compute conformally covariant actions and operators for tensors with mixed symmetries in arbitrary dimension $d$. Our results complete the classification of conformal actions that are quadratic on arbitrary tensors with three indices,…
In this paper we study generic M(atrix) theory compactifications that are specified by a set of quotient conditions. A procedure is proposed, which both associates an algebra to each compactification and leads deductively to general…
Transformers are deep neural network architectures that underpin the recent successes of large language models. Unlike more classical architectures that can be viewed as point-to-point maps, a Transformer acts as a measure-to-measure map…
The definition of symbolic descriptions that consistently represent relevant geometrical aspects in manipulation tasks is a challenging problem that has received little attention in the robotic community. This definition is usually done…
The matrix expressions for every parts of a transformer are firstly described. Based on semi-tensor product (STP) of matrices the hypervectors are reconsidered and the linear transformation over hypervectors is constructed by using…