Ivan Kyrchei

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics · Differential equations and theory of functions

Determinantal representation of the Moore–Penrose inverse over the quaternion skew field is obtained within the framework of a theory of the column and row determinants. Using the obtained analogues of the adjoint matrix, we get the Cramer rules for the least squares solution of left and right systems of quaternionic linear equations.

New definitions of determinant functionals over the quaternion skew field are given in this paper. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer's rules for right and left quaternionic systems of linear equations have been obtained.

The least squares solutions with the minimum norm of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are considered in this paper. We use the determinantal representations of the Moore - Penrose inverse obtained earlier by t...

This paper studies new characterizations and expressions of the weak group (WG) inverse and its dual over the quaternion skew field. We introduce a dual to the weak group inverse for the first time in literature and give some new characterizations for both the WG inverse and its dual, named the right and left weak group inverses for quaternion matr...

The different systems of Sylvester quaternion matrix equations have prolific functions in system and control. This paper considers a Hermitian solution of a system of Sylvester quaternion matrix equations over a quaternion algebra H. If some necessary and sufficient conditions are fulfilled, the general solution to these quaternion matrix equations...

The notions of the Drazin-star and star-Drazin matrices are expanded to quaternion matrices in this paper. Their determinantal representations are developed in both cases in terms of noncommutative row-column determinants of quaternion matrices and for minors of appropriate complex matrices. We study all possible two-sided quaternion matrix equatio...

This article explores Sylvester quaternion matrix equations and potential applications, which are important in fields such as control theory, graphics, sensitivity analysis, and three-dimensional rotations. Recognizing that the determination of solutions and computational methods for these equations is evolving, our study contributes to the area by...

In this paper, the quaternion differential equation, Ax′(t)+Bx(t)=f(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{A}}{\textbf{x}}'(t)+ {\textbf{B}}{\textbf...

A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, and the opposite orientation is assigned the inverse of this quaternion unit. In this paper, we provide a combinatorial description of the determinant of the Laplacian matrix of a quaternion unit gain graph by using row-column noncommutative determ...

A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, and the opposite orientation is assigned the inverse of this quaternion unit. In this paper, we provide a combinatorial description of the determinant of the Laplacian matrix of a quaternion unit gain graph by using row-column noncommutative determ...

The main goal of this paper is to solve new quaternion matrix equations with constrains and present their solution. It is proved that these solutions are expressions that involve adequate weighted generalized inverses. We consider generalized Cramer’s-type representations for derived solutions to new quaternion matrix equations under specified rest...

Abstract In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing $$*$$ ∗ -hermicity over complex field. The compact formula of the general solution of this system is presented in terms of generalized inverses when some necessary and sufficient conditions are fulfilled. An algorithm and a num...

Inverse Problems - Recent Advances and Applications examines some recent advances in inverse problems, new aspects of their mathematical modeling of inverse problems regarding in relation to their applications in physical systems and used the computational methods used. It consists of five chapters divided into two sections. Section 1, “Modeling an...

This chapter is devoted to the survey of quaternion restricted two-sided matrix equation AXB = D and approximation problems related with it. Unique solutions to the considered approximation matrix problems and the restricted quaternion two-sided matrix equations with specific constraints are expressed in terms of the core-EP inverse and the dual co...

This chapter is devoted to \textcolor{red}{the survey of} quaternion restricted two-sided matrix equation $\mathbf{AXB}=\mathbf{D}$ and approximation problems related with it. Unique solutions to the considered approximation matrix problems and the restricted quaternion two-sided matrix equations with specific constraints are expressed in terms of...

Keeping in view that a lot of physical systems with inverse problems can be written by matrix equations, the least-norm of the solution to a general Sylvester matrix equation with restrictions A1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4,A3X1B3+A4X2B4=Cc, is researched in this chapter. A novel expression of the general solution to this system is established and...

The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations fo...

Keeping in view the latest development of anti-Hermitian matrix in mind, we construct some closed form formula for a classical system of matrix equations having anti-Hermitian nature in this paper. We give the necessary and sufficient conditions for the existence of its solution by applying the properties of matrix rank. The general solution to thi...

Sylvester-like matrix equations are encountered in many areas of control engineering and applied mathematics. In this paper, we construct some necessary and sufficient conditions for the system of Hermitian mixed type generalized Sylvester matrix equations to have a solution. The closed form formula to compute the general solution is also establish...

This research aims to introduce and investigate the right and left W-MPCEP, W-CEPMP, and W-MPCEPMP generalized inverses for quaternion matrices. These generalized inverses are introduced as extensions of corresponding generalized inverses applicable to complex matrices. Some new characterizations and expressions of these inverses are presented. Det...

A generalized inverse of a matrix is an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses exist for an arbitrary matrix and coincide with a regular inverse for invertible matrices. The most famous generalized inverses are the Moore–Penrose inverse and the Drazin inverse. Recently, new generalized inv...

The notions of the MPCEP inverse and ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}CEPMP inverse are expanded to quaternion matrices and their deter...

ESI Annual Report 2022. The Erwin Schrodinger International Institute For Mathematics And Physics (ESI), founded in 1993 and part of the University of Vienna since 2011, is dedicated to the advancement of scholarly research in all areas of mathematics and physics and, in particular, to the promotion of exchange between these disciplines. In order t...

Generalized Inverses: Algorithms and Applications demonstrates some of the latest hot topics on generalized inverse matrices and their applications. Each article has been carefully selected to present substantial research results. Topics discussed herein include recent advances in exploring of generalizations of the core inverse, particularly in co...

An inverse of a square matrix can be written by using the cofactor matrix that gives its determinantal representations and inducts Cramer's rule for systems of linear equations with invertible coefficient matrices. Though a computational significance of Cramer's rule is not efficient for systems with a lot of linear equations, it has a large theore...

We introduce three kinds of new weighted quaternion-matrix minimization problems in order to extend some well-known constrained approximation problems. The main result is the claim that these new minimization problems have unique solutions which are expressed in terms of expressions involving weighted core-EP inverse and its dual for adequate quate...

We consider the solvability of four new restricted quaternion matrix equations (QME) and prove that these equations have the unique solutions determined by adequate MPD and DMP inverses. Several particular cases of these equations are presented too. Determinantal representations of solutions to new constrained equations and their particular cases a...

New properties and representations for the core-EP inverse are developed. Particularly, the core-EP inverse of an upper triangular matrix and its sign pattern are considered. Determi- nantal representations for the core-EP inverse and core-EP solution of linear systems are investigated. Corresponding representations of the weighted core-EP inverse...

Generalized Sylvester quaternion matrix equation with some constrictions is explored in this paper. A novel expression of the general solution of the constraint system is given with some necessary and sufficient conditions. Its Cramer’s rule is derived within the framework of the theory of row-column quaternion determinants. An example is given to...

Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: min∥AXB−C∥Fmin‖AXB−C‖F subject to the constraints imposed to the right column space of AA and the left row space of BB. Unique solution to the considered Q-matrix problem is...

Some necessary and sufficient conditions for the existence of the ?-skew-Hermitian solution quaternion matrix equations the system of matrix equations with ?-skew-Hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X = -X?*; Y=-Y?*, A3XA?*3 + B3YB?*3=C5, are established in this paper by using rank equalities of the coefficient matrices. The general...

Our contribution is the development of novel representations and investigations of main properties of the MPCEP inverse. Precisely, we present representations of the MPCEP inverse which involve appropriate Moore–Penrose inverses, projections and full-rank decompositions, as well as limit and integral representations. Determinantal representations f...

In this paper, we extend notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the CMP inverse to matrices over the quaternion skew field that have some features in comparison to these inverses over the complex field. We give the direct methods of their computing, namely, their determinantal representations...

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. Systems of linear equations with several unknowns are naturally represented using the formalism of matrices and vectors. So we arrive at the matrix algebra, etc. Linear algebra is central to almost all areas of mathematics. Many ideas and m...

In this paper, new notions of the weighted core-EP left inverse and the weighted MPD inverse which are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively, are introduced and represented. The direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinant...

By using the determinantal representations of the Moore–Penrose inverse matrix, within the framework of the theory of noncommutative column–row determinants, we obtain determinantal representations (analogs of the Cramer rule) for the solution of the generalized Sylvester quaternion matrix equation AXB + CYD = E .

In this paper, we extend notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the CMP inverse to matrices over the quaternion skew field H that have some features in comparison to these inverses over the complex field. We give the direct methods of their computing, namely, their determinantal representatio...

Generalized inverse matrices are important objects in matrix theory. In particular, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP...

In this paper, we extend notions of the core inverse, core-EP inverse, DMP inverse, and CMP inverse over the quaternion skew field \({\mathbb {H}}\) that have some features in comparison to complex matrices. We give the direct method of their computing, namely, their determinantal representations by using column and row noncommutative determinants...

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Mo...

We constitute some necessary and sufficient conditions for the system A1X1=C1 , X1B1=C2 , A2X2=C3 , X2B2=C4 , A3X1B3+A4X2B4=Cc , to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also rese...

We determine some necessary and sufficient conditions for the existence of the η -skew-Hermitian solution to the following system AX-(AX)η⁎+BYBη⁎+CZCη⁎=D,Y=-Yη⁎,Z=-Zη⁎ over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we...

The system of two-sided quaternion matrix equations with η-Hermicity, A1XA1η* = C1 A 1 X A 1 η * = C 1 $ {\mathbf{A}}_1\mathbf{X}{\mathbf{A}}_1^{\eta \mathrm{*}}={\mathbf{C}}_1$ , A2XA2η* = C2 A 2 X A 2 η * = C 2 $ {\mathbf{A}}_2\mathbf{X}{\mathbf{A}}_2^{\eta \mathrm{*}}={\mathbf{C}}_2$ is considered in the paper. Using no...

Using determinantal representations of the Moore-Penrose inverse previously obtained by the author within the framework of the theory of quaternion column-row determinants, we first get explicit determinantal representation formulas (analogs of Cramer’s rule) of \(\eta \)-Hermitian and \(\eta \)-skew-Hermitian solutions to the quaternion matrix equ...

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer's rule) of a solution to the system of two-sided quaternion matrix equations A1XB1=C1, A2XB2=C2 and its special cases with one one-sided equation when B1=I or A1=I, where I is an id...

In this paper we extend notions of the core inverse, core EP inverse, DMP inverse, and CMP inverse over the quaternion skew-field ${\mathbb{H}}$ and get their determinantal representations within the framework of the theory of column-row determinants previously introduced by the author. Since the Moore-Penrose inverse and the Drazin inverse are nec...

In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving ⁎ -Hermicity AXA⁎+BYB⁎=C over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.

A basic theory on the first order right and left linear quaternion differential systems (LQDS) is given systematic in this paper. To proceed the theory of LQDS we adopt the theory of column-row determinants recently introduced by the author. In this paper, the algebraic structure of their general solutions are established. Determinantal representat...

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer’s rule) of solutions and Hermitian solutions to the system of two-sided quaternion matrix equations A1XA1⁎=C1 and A2XA2⁎=C2 . Since the Moore-Penrose inverse is a necessary tool to...

Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore–Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer’s Rule) to the quaternion two-sided generalized Sylvester matrix equation \( \mathbf{A}...

Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore-Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer's rule) to the quaternion two-sided generalized Sylvester matrix equation $ {\bf A}_{1}...

In 1920, E. H. Moore formulatedformulated the generalized inverse of a matrix in an algebraic setting. e purpose of constructing a generalized inverse matrix is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible matrices. In 1955, R. Penrose showed that the Moore "reciprocal inverse" could b...

Within the framework of the theory of quaternion column–row determinants and using determinantal representations of the Moore–Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer’s rule) to the systems of quaternion matrix equations \( \mathbf{A}_{1}{} \mathbf{X}=\m...

The theory of noncommutative column-row determinants (previously introduced by the author) is extended to determinantal representations of the weighted Moore-Penrose inverse over the quaternion skew field in the chapter. To begin with, we introduce the weighted singular value decomposition (WSVD) of a quaternion matrix. Similarly as the singular va...

formulated the generalized inverse of a matrix in an algebraic setting. e purpose of constructing a generalized inverse matrix is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible matrices. In , R. Penrose showed that the Moore "reciprocal inverse" could be represented by four equations, no...

Weighted singular value decomposition of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore–Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternion...

Weighted singular value decomposition (WSVD) and a representation of the weighted Moore–Penrose inverse of a quaternion matrix by WSVD have been derived. Using this representation, limit and determinantal representations of the weighted Moore–Penrose inverse of a quaternion matrix have been obtained within the framework of the theory of noncommutat...

In this paper, using row-column determinants previously introduced by the author, properties of the determinant of a Hermitian matrix are investigated, and determinantal representations of the inverse of a Hermitian coquaternionic matrix are given. With their help, Cramer’s rules for left and right systems of linear equations with Hermitian coquate...

Weighted singular value decomposition (WSVD) of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore-Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted qua...

A generalized inverse of a given quaternion matrix (similarly, as for complex matrices) exists for a larger class of matrices than the invertible matrices. It has some of the properties of the usual inverse, and agrees with the inverse when a given matrix happens to be invertible. There exist many different generalized inverses. In this chapter, we...

Matrix equations play a fundamental role in the matrix theory and other elds of linear algebra and matrix analysis. Since applications of matrix equations are found in most scientiic elds, then they arouse interest, both in the theoretical and in applied components, which make this subject an important topic in the mathematical research. In the rec...

In this paper properties of the determinant of a Hermitian matrix are investigated, and determinantal representations of the inverse of a Hermitian coquaternionic matrix are given. By their using, Cramer's rules for left and right systems of linear equations with Hermitian coquaternionic matrices of coefficients are obtained. Cramer's rule for a tw...

Within the framework of the theory of the noncommutative columnrow determinants previously introduced by the author and using obtained quaternion weighted singular value decomposition we get determinantal representations of the weighted Moore-Penrose inverse of an arbitrary quaternion matrix.

Weighted singular value decomposition (WSVD) and a representation of the weighted Moore-Penrose inverse of a quaternion matrix by WSVD have been derived. Using this representation, limit and determinantal representations of the weighted Moore-Penrose inverse of a quaternion matrix have been obtained within the framework of the theory of the noncomm...

By using determinantal representations of the W -weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the W -weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equations W A W...

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider...

By using determinantal representations of the W-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the W-weighted Drazin inverse solutions (analogs of Cramer's rule) of the quaternion matrix equations $ {\bf W}{\bf...

Since product of quaternions is noncommutative, there is a problem how to determine a determinant of a matrix with noncommutative elements (it's called a noncommutative determinant). We consider two approaches to define a noncommutative determinant. Primarily, there are row -- column determinants that are an extension of the classical definition of...

This book presents original studies on the leading edge of linear algebra. Each chapter has been carefully selected in an attempt to present substantial research results across a broad spectrum. The main goal of Chapter One is to define and investigate the restricted generalized inverses corresponding to minimization of constrained quadratic form....

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider...

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider...

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Book of abstracts of the International Algebraic Conference dedicated to 100tr anniversary of L.

Within the framework of the theory of the column and row determinants, we obtain determinantal representations of the Drazin inverse both for Hermitian and arbitrary matrices over the quaternion skew field. Using the obtained determinantal representations of the Drazin inverse we get explicit representation formulas (analogs of Cramer’s rule) for t...

The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column and row immanants (permanents) for matrices over a non-split quaternion algebra are introduced, and their bas...

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Within the framework of the theory of the column and row determinants, we obtain determinantal representations of the Drazin inverse for Hermitian matrix over the quaternion skew field. Using the obtained determinantal representations of the Drazin inverse we get explicit representation formulas (analogs of Cramer's rule) for the Drazin inverse sol...

The Drazin inverse solutions of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are considered in this paper. We use both the determinantal representations of the Drazin inverse obtained earlier by the author and in the pape...

Within the framework of the theory of the column and row determinants, we obtain explicit representation formulas (analogs of Cramer's rule) for the minimum norm least squares solutions of quaternion matrix equations ${\bf A} {\bf X} = {\bf B}$, $ {\bf X} {\bf A} = {\bf B}$ and ${\bf A} {\bf X} {\bf B} = {\bf D} $.

Within the framework of the theory of column and row determinants, we have obtained the determinantal representation of the Moore–Penrose inverse matrix over the quaternion skew field.

New definition of determinant functionals (the column and row determinants) over the quaternion division algebra are given in this chapter. We study their proper- ties and relations with other well-known noncommutative determinants (Study,Moore, Diedonne, Chen) and the quasideterminants of Gelfand-Retakh. We introduce a defi nition of a determinant...

The least squares solutions with the minimum norm of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are considered in this paper. We use the determinantal representations of the Moore - Penrose inverse obtained earlier by t...

Explicit representation formulas for the least squares solution of the quaternion matrix equation AXB=C

In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.