Related papers: Natural Product Xn on matrices
This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc.…
The material of this work is aimed at mathematics educators, as well as math specialists with a keen interest in progressions. In this paper, we study the subject of arithmetic, geometric, mixed, and harmonic progressions or sequences. Some…
In recent years, conventional chemistry techniques have faced significant challenges due to their inherent limitations, struggling to cope with the increasing complexity and volume of data generated in contemporary research endeavors.…
Section I contains introductory remarks about surface motions. Section II gives a detailed derivation of $H=-\Delta-Tr\sum_{i<j}[X_i,X_j]^2$ as describing a quantized discrete analogue of relativistically invariant membrane dynamics.…
In this book i treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a…
Let K denote a field. Given an arbitrary linear subspace V of M_n(K) of codimension lesser than n-1, a classical result states that V generates the K-algebra M_n(K). Here, we strengthen this in three ways: we show that M_n(K) is spanned by…
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…
Let F(n) be a polynomial of degree at least 2 with integer coefficients. We consider the products N_x=\prod_{1 \le n \le x} F(n) and show that N_x should only rarely be a perfect power. In particular, the number of x \le X for which N_x is…
Matrices over the dual numbers are considered. We propose an approach to classify these matrices up to similarity. Some preliminary results on the realization of this approach are obtained. In particular, we produce explicitly canonical…
These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.
Malcev described the congruences of the monoid $T_n$ of all full transformations on a finite set $X_n=\{1, \dots,n\}$. Since then, congruences have been characterized in various other monoids of (partial) transformations on $X_n$, such as…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We prove a general result on presentations of finitely-generated algebras and apply it to obtain nice presentations for some noncommutative algebras arising in the matrix bispectral problem. By "nice presentation" we mean a presentation…
Supertropical matrix theory was investigated in [6], whose terminology we follow. In this work we investigate eigenvalues, characteristic polynomials and coefficients of characteristic polynomials of supertropical matrices and their powers,…
This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by…
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
We take matrix decompositions that are usually applied to matrices over the real numbers or complex numbers, and extend them to matrices over an algebra called the double numbers. In doing so, we unify some matrix decompositions: For…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…