Some Investigations On Generalized Inverses Of Matrices Over Semirings And Quaternion Matrix Equations

A generalized inverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. Solving linear equations, choosing a particular solution (e.g. least norm) if the equation has more than one, or producing an approximate solution (e.g. least squares solution) if the equation does not have any, are some of the useful properties of generalized inverses.The concept of the generalized inverse seems to have been first mentioned in print in 1903 by Fredholm [1], where a particular generalized inverse of an integral operator was given. The class of all generalized inverses was characterized in 1912 by Hurwitz [2]. The algebraic nature of generalized inverses of matrices was established in the works of Moore [3], Penrose [4], Drazin [5] and others. In 1955, Penrose [4] showed that, for any finite matrix A over C (R), there exist a unique matrix X satisfying the following four equations Because this unique generalized inverse had been studied earlier in 1920 by Moore [3] in a different way, it is commonly known as the Moore-Penrose inverse, and is denoted by A^. Above mentioned four equations are known as Penrose equations. Note that if A is nonsingular than X=A-1 trivially satisfies the Penrose equations which shows that the Moore-Penrose inverse of a nonsingular matrix is same as the ordinary inverse. The generalized inverses which satisfy some, but not all, of the four Penrose equations are also of great importance. In particular, any matrix X satisfying the first Penrose equation (1) is called a{1} inverse or inner inverse of A, we shall denote an inner inverse of A by A-. Any matrix X satisfying simultaneously the Penrose equations (1) and (2), is called a{1,2} inverse or reflexive inverse of A, we shall denote a reflexive inverse of A by A+.Some other generalized inverses which have important spectral properties i.e., the properties related to the eigenvalues and eigenvectors of matrices, include the Drazin inverse and the group inverse. Let A∈Cnxn and Ind(A)= k. Then the matrix is called the Drazin inverse of A. If k=1, then X is called the group inverse of A.Thousands of articles appeared on this topic, since Penrose's article [4]. The results on theory and applications of generalized inverses of real or complex matrices can be found in many well-known monographs, see for instance [36]-[38].However, in the last three or four decades, the range of mathematical models has been considerably enlarged as a result of the development of new modeling paradigms such as fuzzy set theory [50]-[54] and also due to the needs for new problem-solving techniques on graphs [55]-[57], and new operation research and control theory ap-plications. From these recent evolutions, a large number of researchers studied the generalized inverses of matrices, such as the Moore-Penrose inverse, the group inverse and the Drazin inverse in more general algebraic settings, such as commutative rings [26], arbitrary rings [27]-[29] and idempotent semirings [30]. This is a motivation for our research to investigate the generalized inverses for matrices over an arbitrary semiring.A semiring consists of a set S and two binary operations on S, addition (+) and multiplication (-), such that:(1) (S,+) is an Abelian monoid (identity denoted by 0);(2) (S,. ) is a monoid (identity denoted by 1);(3) multiplication distributes over addition from either side;(4) s0=0s=0 for all s∈S; and(5) 1≠0.According to the definition of semiring, the algebraic structures such as the max-plus algebra or minrnax algebra in ([82], [83]), dioid in ([68], [69]). fuzzy algebra in [84], incline algebra in [85], bottleneck algebra in [86], all rings with identity, and the nonnegative real numbers with usual addition and multiplications are some examples of semirings.This dissertation is composed of four chapters.Chapter 1, presents some basic knowledge on the theory of generalized inverses of matrices and its applications in solutions of linear matrix equations over the field of complex numbers. This Chapter is mainly divided into two sections, in first Section, we discuss the existence and some basic important properties of various kinds of generalized inverses of matrices over the field of complex numbers. We shall focus on the Moore-Penrose inverse, the Drazin inverse and the group inverse of matrices In second section, the necessary and sufficient conditions for the existence and the expressions of the general solutions to some classical linear matrix equations using the generalized inverses of matrices will be discussed. This first Section of this chapter will provide a motivation and background knowledge for chapters 2 and 3. The second Section is background knowledge for Chapter 4.In Chapter 2, we discuss the generalized invertibility of a regular matrix over an arbitrary semiring. Basically, we investigate the following two questions in this Chapter;1. What are the necessary and sufficient conditions for the existence of the Moore-Penrose inverse, the group inverse and the group-Moore-Penrose inverse of matrix A over an arbitrary semiring S?2. Can we establish some explicit expressions for these generalized inverses?We have successfully established some necessary and sufficient conditions for the existence of the group inverse, the Moore-Penrose inverse and the group-Moore-Penrose inverse of a regular matrix over an arbitrary semiring. Some explicit expres-sions of these generalized inverses are also presented. We have used pure algebraic techniques to establish our results because most of fruitful techniques of matrices over field such as, matrix decompositions and useful properties of matrix rank are not helpful in general settings of semirings.Chapter 3 is concerned with the Drazin invertibility for the elements of an arbi-trary semiring. It is well-known that the Drazin inverse for matrices is defined for square matrices and the set Mn(S) of all n x n matrices over an arbitrary semiring S is also a semiring under usual operations of addition and multiplication of matri-ces, therefore, it is meaningful to investigate the drazin invertibility in more general algebraic settings; for the elements of an arbitrary semiring.. In this Chapter, We establish the necessary and sufficient conditions for the existence and explicit expres-sions of the Drazin inverse of an element in an arbitrary semiring. Moreover, we consider the product paq under some additional necessary conditions for which the Drazin inverse of the product paq exists. The results of this Chapter are obviously true for matrices over an arbitrary semiring. As a special case, the group inverse for elements of semiring is also discussed.One of the most important properties of the generalized inverse matrices is that, these are very useful in solving linear matrix equations and establishing explicit ex-pressions of the general solutions of linear matrix equations. In Chapter 4 of this dissertation, we discuss solutions of some linear matrix equations over the skew field of quaternions by using generalized inverse matrices and minimal ranks of matrix ex-pressions, due to the recent applications of quaternion matrices in computer graphics, signal processing, quantum physics, altitude control, and mechanics ([109]-[114]).Quaternions are vectors in 4-dimensions endowed with a rule for multiplication that is associative but not commutative, distributes through addition, contains an identity, and, most crucially, for which each nonzero vector in 4-dimensions has a unique multiplicative inverse. Thus the quaternions form a division algebra usu-ally denoted by H The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in [107]. As an area of mathematics (algebra, analysis, geometry, and computation; see, e.g., [108]-[111]), quaternions have been extensively studied. Shoemake [112] popularized quaternions in the world of computer graphics. In computer graphics, quaternions can be used to reduce storage and to speed up calculations involving rotations. A quaternion is represented by just four scalars, in contrast to a 3 x 3 rotation matrix which has nine scalar entries.In Chapter 4, we mainly investigate the minimal rank of quaternion matrix ex-pression with respect to X and Y= Y(*); and the minimal rank of quaternion matrix expres-sion with respect to X and Y=-Y(*). To establish the minimal rank formulas for these matrix expressions, we use generalized inverse matrices and some basic rank equalities of block matrices. As applications, we study the following quaternion matrix equationswe establish the necessary and sufficient conditions for the existence of the general solutions, the persymmetric solutions, and the perskewsymmetric solutions to these linear quaternion matrix equations. Moreover. the expressions of the corresponding solutions to the matrix equations are also given when the solvability conditions are satisfied.