A NOTE ON EIGENVALUES OF BICOMPLEX MATRIX (original) (raw)
DIAGONALIZATION OF THE BICOMPLEX MATRIX
International Journal of Research Publication and Reviews (IJRPR), 2022
In this paper, we have studied and investigated the properties of bicomplex matrices. On invertible and non-invertible bicomplex matrices, we have obtained certain results. The bicomplex matrix's eigenvalues, eigenvectors and spectrum are also introduced. For bicomplex matrices, the Cayley-Hamilton theorem has been investigated. Furthermore, we discussed the Diagonalizable Bicomplex matrix along with certain results.
On Some Properties of Determinants of Bicomplex Matrices
In this paper, we have studied Determinants of Bicomplex Matrices and investigated their properties. We have introduced Bicomplex Symmetric Matrix, Bicomplex Skew-Symmetric Matrix, Bicomplex idempotent matrix, Bicomplex Skew-idempotent matrix, Bicomplex Involutory matrix, Bicomplex skew-involutory matrix, three types of Hermetian and Skew-Hermetian Matrix, Bicomplex Orthogonal Matrix and three types of Unitary Matrix and investigated the properties of their determinants.
ON SOME PROPERTIES OF DETERMINANTS OF BICOMPLEX MATRICES JSTR
Journal of Science and Technological Researches, 2022
In this paper, we have studied determinants of bicomplex matrices and investigated their properties. We have introduced bicomplex symmetric matrix, bicomplex skew-symmetric matrix, bicomplex idempotent matrix, bicomplex skew-idempotent matrix, bicomplex involutory matrix, bicomplex skew-involutory matrix, three types of Hermetian and skew-Hermetian matrix, bicomplex orthogonal matrix and three types of unitary matrix and investigated the properties of their determinants. 2010 AMS Subject Classification. 30G35, 32A30, 32A10, 15A15, 15B57, 15B10, 15A18. Keywords: Bicomplex matrix, determinant of bicomplex matrix, orthogonal matrix, unitary matrix.
Certain Results on Bicomplex Matrices By Anjali & Amita
Global Journal of Science Frontier Research, 2018
This paper begins the study of bicomplex matrices. In this paper, we have defined bicomplex matrices, determinant of a bicomplex square matrix and singular and non-singular matrices in C 2. We have proved that the set of all bicomplex square matrices of order n is an algebra. We have given some definitions and results regarding adjoint and inverse of a matrix in C 2. We have defined three types of conjugates and three types of tranjugates of a bicomplex matrix. With the help of these conjugates and tranjugates, we have also defined symmetric and skew-symmetric matrices, Hermitian and Skew-Hermitian matrices in C 2 .
On Some Matrix Representations of Bicomplex Numbers
2019
In this work, we have defined bicomplex numbers whose coefficients are from the Fibonacci sequence. We examined the matrix representations and algebraic properties of these numbers. We also computed the eigenvalues and eigenvectors of these particular matrices.
Algebraic and geometric characteristics of bicomplex variables
2017
The notion of bicomplex variable appears at the end of XIX century, when it was nominated by Corado Segre . Today, one can to see the names of many authors, which had studied bicomplex analysis, cited in the book of G. Baley-Price . We recall too the paper of Stefan Röon, who remark the interconnection of bicomplex study and Fueter's regular function (to see Helsinki Technological University Press [3]). The porpose of our paper is to extend the complex treatment of complex analysis to the Gelfand's complex algebras theory. Let us recall that G. Baley-Price considers only multicomplex spaces.
Bicomplex Version of Cayley Hamilton Theorem
Theory of matrices is an integral part of algebra as well as Theory of equations. Matrices plays an important role in every branch of Physics, Computer Graphics and are also used in representing the real world's data and there are so many applications of matrices ,for this reason we thought of studying bicomplex matrices. The monograph by Price [4] contains few exercises pertaining to matrices with bicomplex entries. In this paper we discussed the bicomplex version of CAYLEY – HAMILTON THEOREM and also used it to evaluate the inverse of a non singular bicomplex matrix.
A note on Bicomplex Linear operators on bicomplex Hilbert spaces
International Journal of Mathematics Trends and Technology, 2016
In this paper we define the isomorphism between the bicomplex Hilbert spaces. We also give some simple and basic results on bicomplex isomorphism with respect to hyperbolic-valued norm on the bicomplex Hilbert spaces.
A New Approach to Bicomplex Jacobsthal Matrix and Bicomplex Jacobsthal-Lucas Matrix Components
Journal of Advances in Mathematics and Computer Science
In this present paper, we give a detailed study of a new generation of Bicomplex Jacobsthal matrix and Bicomplex Jacobsthal-Lucas matrix using Jacobsthal -matrix and Jacobsthal Lucas -matrix. Also presented some formulas, facts, and properties about these matrices. In addition, a new vector called the bicomple Jacobsthalvectors and the bicomplex Jacobsthal-Lucas vectors with matrix components is presented. Some properties of this vector apply to various properties of geometry which are not generally known in the geometry of space. Then, using this matrix representation, we give some identities.
The Real Matrices forms of the Bicomplex Numbers and Homothetic Exponential motions
2014
In this paper, a bicomplex number is described in four- dimensional space and its a variety of algebraic properties is presented. In addition, Pauli-spin matrix elements corresponding to base the real matrices forms of the bicomplex numbers are obtained and its the algebraic properties are given. Like i and j in two different spaces are defined terms of Euler's formula. In the last section velocities become higher order by giving an exponential homothetic motion for the bicomplex numbers. And then, Due to the way in which the matter is presented, the paper gives some formula and facts about exponential homothetic motions which are not generally known
On Dual Bicomplex Numbers and Their Some Algebraic Properties
2019
The object of this work is to contribute to the development of bicomplex numbers. For this purpose, in this study we firstly introduced bicomplex numbers with coefficients from complex Fibonacci sequence. And then, using Babadag's work [1], we examined the dual form of the newly defined numbers. Moreover, we gave some fundamental identities such as Cassini and Catalan identities provided by the elements in defined sequence.
On the Spectra of Some Matrices Derived from Two Quadratic Matrices
2016
Abstract. The relations between the spectrum of the matrix Q+ R and the spectra of matrices (γ+δ)Q+(α+β)R−QR−RQ, QR− RQ, αβR−QRQ, αRQR − (QR)2, and βR−QR have been given assuming that the matrix Q+R is diagonalizable, where Q, R are {α, β}-quadratic matrix and {γ, δ}-quadratic matrix, respectively, of order n. 1.
Some Algebraic and Analytical Properties of Special Matrices
AL-Rafidain Journal of Computer Sciences and Mathematics, 2011
In this work we present a subset of M is the set of all n n matrices ) which we called the set of special matrices and denoted it by n n S . We give some important properties of n n S .
On the eigenvalues of matrices with given upper triangular part
Integral Equations and Operator Theory, 1990
We give necessary and sufficient conditions for a set of numbers to be the eigenvalues of a completion of a matrix prescribed in its upper triangular part. 1. INTRODUCTION. Let F be a given part of entries of an n• matrix A=[ai/]~,j=! over a field F. The problem of describing the eigenvalues of A when the rest of the entries a i j~ [" take arbi-Irary values in F is called an eigenvalue completion problem. A number of such problems, sometimes including in addition certain characteristics of the eigenvalues (such as multiplicities), have been solved in the last 10-20 years. Here is a partial list of related papers: For the case when F is a principal block see Thompson [T] and de Sh [dS]. The cases when F is a full width (or length) block was studied by Rosenbrock [R] and Zaballa [ZI]. Off-diagonal block F was solved by Gohberg, Kaashoek and van Schagen [GKS] and Zaballa [Z2]. For the case of diagonal F see Mirsky [M] and also Johnson and Shapiro [JS]. Finally we mention the case of two complementary principal blocks which was studied by De Oliveira [D1,D2] and Silva IS1] who studied the case of two complementary non-principal blocks as well [$2]. Here we solve the eigenvalue completion problem for a new case when F is the upper triangular part of A including the diagonal. This result allows us to compute the smallest spectral radius among all completions. It may be viewed as a solution of a finite dimensional analogue of a Nehari type spectral interpolation problem discussed recently by H. Bercovici, C. Foias and A. Tannenbanm [BFT1,BVI'2]. These papers were an inspiration for the authors.
The word "matrix" comes from the Latin word for "womb" because of the way that the matrix acts as a womb for the data that it holds. The first known example of the use matrices was found in a Chinese text called Nine Chapters of the Mathematical Art, which is thought to have originated somewhere between 300 B.C. and 200 A.D. The modern method of matrix solution was developed by a German mathematician and scientist Carl Friedrich Gauss. There are many different types of matrices used in different modern career fields. We introduce and discuss the different types of matrices that play important roles in various fields.
On eigenvalues of rectangular matrices
2007
Given a (k+1)(k+1)(k+1)-tuple A,B1,...,BkA, B_1,...,B_kA,B1,...,Bk of (mtimesn)(m\times n)(mtimesn)-matrices with mlenm\le nmlen we call the set of all kkk-tuples of complex numbers la1,...,lak\{\la_1,...,\la_k\}la1,...,lak such that the linear combination A+la1B1+la2B2+...+lakBkA+\la_1B_1+\la_2B_2+...+\la_kB_kA+la_1B_1+la2B2+...+lakBk has rank smaller than mmm the {\it eigenvalue locus} of the latter pencil. Motivated primarily by applications to multi-parameter generalizations of the Heine-Stieltjes spectral problem, see \cite{He} and \cite{Vol}, we study a number of properties of the eigenvalue locus in the most important case k=n−m+1k=n-m+1k=n−m+1.
On Some Properties of Bicomplex Numbers •Conjugates • Inverse •Modulii
We have done some work on Bicomplex numbers. Some glimpses of the work can be seen in these papers [6, 7, 8, 9, 10, 11, 12, 13, 14]. In the present paper an attempt has been made to discuss and establish extensive algebraic properties of Bicomplex Numbers. Properties of three types of conjugation of Bicomplex Numbers have been determined and established some relation between them. Invertible and Non – invertible Bicomplex numbers have been investigated and findings are discussed. Properties of Modulii of Bicomplex numbers have been shown along with some relation between them.
On Copositive Matrices and their Spectrum
2006
Let A 2 R n n . We provide a block characterization of copositive matrices, with the assumption that one of the principal blocks is positive denite. Haynsworth and Homan showed that if r is the largest eigenvalue of a copositive matrix then r j j, for all other eigenvalues of A. We continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues. Moreover, if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues, then it is possible to increase the the nonnegative eigenvalues to form a copositive matrix A 0 , without changing the eigenvectors. We also show that if a copositive matrix has just one positive eigenvalue, and n 1 nonpositive eigenvalues then A has a nonnegative eigenvector corresponding to a nonnegative eigenvalue.