arXiv subjects

Akhil Prakash

Publications and source records attributed to Akhil Prakash.

6 recordsLinked to original sources

arXiv2026

Inner Products and Banach Algebra structures on Bicomplex Numbers and Their Associated Spaces

In this paper, we introduce various types of inner products and norms on the bicomplex number system $\mathbb{C}_2$, the bicomplex vector space ${\mathbb{C}_2}^{n}$, the space of bicomplex matrices ${C_2}^{m \times n}$, and the space of bicomplex polynomials $\mathbb{C}_2[\xi]_n$. We investigate the relationships among these inner products and norms, and establish several results. Furthermore, we prove that $\mathbb{C}_2$ and ${\mathbb{C}_2}^{n}$ are Banach algebras and Hilbert spaces. These results provide a unified framework for the study of inner product structures and normed linear spaces over bicomplex numbers and their associated spaces.

math.RT↗

arXiv2025

Idempotent & Nilpotent Operators and Matrices in Bicomplex Space

This paper explores idempotent and nilpotent operators in bicomplex spaces, focusing on their properties and behavior. We define idempotent and nilpotent matrices in this framework and derive related results. Several theorems are presented to establish conditions for the existence and behavior of bicomplex idempotent and nilpotent operators and bicomplex idempotent matrices.

math.RT↗

arXiv2025

Rank Of bicomplex matrices and system of algebraic equations

In this paper, we study the rank of matrices of bicomplex numbers. The relationship between rank, idempotent column rank and idempotent row rank is examined. Then, the solution of a system of equations in bicomplex space is presented using a new technique. Moreover, we establish a necessary and sufficient condition for the existence of solutions of a system of equations in bicomplex space and derive some related results.

math.RA↗

arXiv2025

A New Perspective on Eigenvalues and Eigenvectors of Bicomplex Linear Operators

This paper deals with eigenvalues and eigenvectors of bicomplex linear operators defined on bicomplex space. We investigate the properties of these operators in the context of eigenvalues and eigenvectors, along with some relevant theorems. Several theorems are explored to establish conditions for bicomplex eigenvalues and eigenvectors. Additionally, we examine the structure of the eigenspaces corresponding to these eigenvalues,analyze their properties, and discuss relevant theorems.

math.RT↗

arXiv2024

On the Rank of a bicomplex matrix

The paper explores the concept of the rank of a bicomplex matrix, delving into four distinct types of ranks and investigating conditions under which these ranks are equivalent. It also defines and analyzes the concept of idempotent row space and idempotent column space of a bicomplex matrix. Some examples and counter examples have been presented to substantiate the study.

math.RA↗

arXiv2023

Matrix representations of linear transformations on bicomplex space

An algebraic investigation on bicomplex numbers is carried out here. Particularly matrices and linear maps defined on them are discussed. A new kind of cartesian product, referred to as an idempotent product, is introduced and studied. The elements of this space are linear maps of a special form. These linear maps are examined with respect to usual notions like kernel, range, and singularity. Their matrix representation is also discussed.

math.RT↗