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Chapter 1 Generalized Inverses Generalized inverses have wide applications in many fields such as numerical analysis�� cryptography�� operations research�� probability statistics�� combinatoric�� optimization�� astronomy�� earth sciences�� managerial economics�� and various engineering sciences [50��56�� 64�� 205]. Because of the different research problems�� there are many types of generalized inverses. For example�� the Moore-Penrose inverse�� {l}-inverse�� {2}-inverse�� Drazin inverse�� group inverse and Bott-Duffin inverse etc. There are several monographs on generalized inverses [5��8��34��60��80��96��104��120��158��164�� 193��194��196�� 200�� 202��207]. Other related research on perturbation analysis and preservation of generalized inverses can be found in [2�� 98��117��186��213��214��224]. This chapter contains some basic concepts and tools necessary to follow the will cover some standard matrix theory tools and various graph nations and invariants. 1.1 Matrix Decompositions In this section�� we list some common matrix decompositions in matrix theory. These decompositions can help readers better understand the properties of generalized inverses of matrices. Let Kmxn�� Cmxn and Rmxn denote the set of all m x n matrices over skew fields K�� complex field C and real field M�� respectively. Let IKn�� Cn and W1 denote the vector set of ?vdimensional over K�� C and R��respectively. Theorem 1.1 [232]. Let A G Kmxn with rank(A) = r. Then there exist invertible matrices P and Q such that where Ir denotes an identity matrix of r-order. The decomposition of theorem 1.1 is called the equivalent decomposition of A. Theorem 1.2 [39�� 232]. Let A G K . Then there exists an invertible matrix P such that where A is an invertible matrix; and iV is a nilpotent matrix. The decomposition of theorem 1.2 is called core-nilpotent decomposition of A. Next�� we introduce the full rank factorization. Theorem 1.3 [8]. Let A G Kmxn with rank(i4) = r. Then there exist a full column rank B G Kmxr and a full row rank C �� Krxn such that A = BC. Proof. Theorem 1.1 gives there exist invertible matrices P and Q such that Take B = P and C = [Ir 0]Q. Then A = BC is a full rank factorization For A G Cmxn�� AT and A* denote the transpose and conjugate transpose of A�� respectively. We know that AA* and A*A are both positive and semidefinite�� and they have the same nonzero eigenvalues of AA*�� where r = rank(A). Then are the singular values of A. Next is the singular value decomposition of A. Theorem 1.4 [94��212��132]. Let A G Cmxn. Then there exist unitary matrices U G Cmxm and V G Cnxn such that where is a diagonal matrix whose diagonal elements are the singular values of A. 1.2 Moore-Penrose Inverse The concept of generalized inverses was first introduced by I. Fredholm [92] in 1903 which is a particular generalized inverse of an integral operator�� called pseudoin-verse. W.A. Hurwitz gave a simple algebraic construction of the pseudoinverse using the finite dimensionality of the null spaces of the Fredholm operators in 1912 [111]. The generalized inverses of differential operators appeared in D. Hilbert��s discussion of the generalized Green��s functions in 1904 [107]. In the Bulletin of the American Mathematical Society�� the generalized inverses of a matrix was first introduced in 1920 by E.H. Moore [156] who is a member of the US National Academy of Sciences�� where a generalized inverse is defined using projectors of matrices. In 1955�� R. Penrose OM FRS�� who is a foreign member of the US National Academy of Sciences and 2020 Nobel Prize in Physics�� gave its equivalent definition using matrix equations [159]. The following definition of generalized inverse is given by R. Penrose. Definition 1.1 [5]. Let A G Cmxn. If a matrix X G Cnxm satisfies the following four equations AXA = A�� XAX = X��(AX)* = AX�� (XA)* = XA�� then X is called the Moore-Penrose inverse of A (abbreviated as the M-P inverse)�� denoted by A +. Obviously�� if is nonsingular�� then. The following theorem will elaborate the M-P inverse uniquely exists. Theorem 1.5 [5]. For a matrix A G Cmxn�� A+ exists and is unique. Proof. By the singular value decomposition of matrices�� there exist unitary matrices and such that ��where is a nonsingular positive diagonal matrix. Next�� it can be verified that X is the M-P inverse of A. Therefore�� A+ always exists. The uniqueness of A + is proved as follows. If X1 and X2 are both the M-P inverse of A�� then Therefore�� A+ exists and is unique. The following theorem is observed from the proof of theorem 1.5.