Related papers: A geometric decomposition of real diagonalizable m…

The general decomposition theory of exponential operators is briefly reviewed. A general scheme to construct independent determining equations for the relevant decomposition parameters is proposed using Lyndon words. Explicit formulas of…

Planar homography, with eight degrees of freedom (DOFs), is fundamental in numerous computer vision tasks. While the positional offsets of four corners are widely adopted (especially in neural network predictions), this parameterization…

In this note, we study a factorization result for graded decomposition maps associated with the specializations of graded algebras. We obtain results previously known only in the ungraded setting.

A uniqueness theorem for an LU decomposition of a totally nonnegative matrix is obtained.

Polytope complexes are the generalisation of polygon meshes in geo-information systems (GIS) to arbitrary dimension, and a natural concept for accessing spatio-temporal information. Complexes of each dimension have a straight-forward…

Let $\mathbb{F}$ be a finite field of odd characteristic. When $|\mathbb{F}|\ge 5$, we prove that every matrix $A$ admits a decomposition into $D+M$ where $D$ is diagonalizable and $M^2=0$. For $\mathbb{F}=\mathbb{F}_3$, we show that such…

We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel…

This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…

It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we…

We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with…

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field $\mathbb{F}_p$. We apply symmetric (row and column) elementary transformations to the adjacency matrix over $\mathbb{F}_p$ in…

A new layers method is presented for multipartite separability of density matrices from simple graphs. Full separability of tripartite states is studied for graphs on degree symmetric premise. The models are generalized to multipartite…

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

We investigate deformations of free and linear free divisors. We introduce a complex similar to the de Rham complex whose cohomology calculates deformation spaces. This cohomology turns out to be zero for many linear free divisors and to be…

In this paper a geometric method based on Grassmann manifolds and matrix Riccati equations to make hermitian matrices diagonal is presented. We call it Riccati Diagonalization.

Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…

In this paper, we will derive the real roots of certain sets of matrices with real entries. We will also demonstrate that real orthogonal matrices can have real root or be involutory. Eventually, we will represent idempotent matrices in a…

We review the enveloping algebra of the 10 dimensional chiral sigma matrices. To facilitate the computation of the product of several chiral sigma matrices we have developed a symbolic program. Using this program one can reduce the…

Let $M=(m_{ij})$ be a symmetric matrix of order $n$ whose elements lie in an arbitrary field $\mathbb{F}$, and let $G$ be the graph with vertex set $\{1,\ldots,n\}$ such that distinct vertices $i$ and $j$ are adjacent if and only if $m_{ij}…

Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic…