Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

New results for EP matrices in indefinite inner product spaces

In this paper we study J-EP matrices, as a generalization of EP-matrices in indefinite inner product spaces, with respect to indefinite matrix product. We give some properties concerning EP and J-EP matrices and find connection between them. Also, we present some results for reverse order law for Moore-Penrose inverse in indefinite setting. Finally, we deal with the star partial ordering and improve some results given in the “EP matrices in indefinite inner product spaces” (2012), by relaxing some conditions.     

Generalized inverses of certain Toeplitz matrices

The spectral inverse A^s of a Toeplitz matrix A whose form is related to that of a circulant matrix is studied by describing the algebraic structure of the semigroup of all matrices commuting with a given matrix with distinct eigenvalues. A computational form for A^s is given and necessary and sufficient conditions are found for A^s to be the Moore-Penrose generalized inverse A⁺.     

Nonnegative mth roots of nonnegative 0-symmetric idempotent matrices

Nonnegative mth roots of nonnegative 0-symmetric idempotent matrices have been characterized. As an application, a characterization of nonnegative matrices A whose Moore-Penrose generalized inverse A² is some power of A is obtained, thus yielding some well-known theorems.     

The generalized weighted Moore-Penrose inverse

In this paper, we definite a generalized weighted Moore-Penrose inverse A _M,N ⁺ of a given matrixA, and give the necessary and sufficient conditions for its existence. We also prove its uniqueness and give a representation of it. In the end we point out this generalized inverse is also a prescribed rangT and null spaceS of {2}-(or outer) inverse ofA.     

On the semi-continuity of generalized inverses in Banach algebras

The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore-Penrose inverse in a C^∗-algebra and therefore, derive some new and well-known criteria in operator theory.     

Reverse Order Law for the Group Inverse in Semigroups and Rings

In this paper, we provide equivalent conditions for the two-sided reverse order law for the group inverse (ab)^# = b ^# a ^# and (ba)^# = a ^# b ^#, in semigroups and rings. Moreover, we prove that, under finiteness conditions, these conditions are also equivalent with the one-sided reverse order law (ab)^# = b ^# a ^#.     

On Euclidean distance matrices

If A is a real symmetric matrix and P is an orthogonal projection onto a hyperplane, then we derive a formula for the Moore-Penrose inverse of PAP. As an application, we obtain a formula for the Moore-Penrose inverse of an Euclidean distance matrix (EDM) which generalizes formulae for the inverse of a EDM in the literature. To an invertible spherical EDM, we associate a Laplacian matrix (which we define as a positive semidefinite n × n matrix of rank n − 1 and with zero row sums) and prove some properties. Known results for distance matrices of trees are derived as special cases. In particular, we obtain a formula due to Graham and Lovász for the inverse of the distance matrix of a tree. It is shown that if D is a nonsingular EDM and L is the associated Laplacian, then D⁻¹ − L is nonsingular and has a nonnegative inverse. Finally, infinitely divisible matrices are constructed using EDMs.     

Full-rank block LDL ? decomposition and the inverses of n×n block matrices

Full-rank block LDL ^? decomposition of a Hermitian n×n block matrix A is examined, where the iterative procedure evaluating the sub-matrices appearing in L and D is provided. This factorization is used to evaluate the inverse and Moore-Penrose inverse of a Hermitian n×n block matrix. The method for the calculation of the Moore-Penrose inverse of an arbitrary 2×2 block matrix is also provided. Therefore, matrix products A ^? A and AA ^? and the corresponding full-rank block LDL ^? factorizations are observed. Also, a simple explicit formulae calculating the solution vector components of the normal system of equations is stated, where the LDL ^? decomposition of the system matrix is done.     

On the covariance of the Moore-Penrose inverse

Given a square complex matrix A with Moore-Penrose inverse A², we describe the class of invertible matrices T such that (TAT^-1)²=TA²T^-1.     

Nonnegativity of principal minors of generalized inverses of P0-matrices

We derive a necessary and sufficient condition under which a reflexive generalized inverse of a singular P₀-matrix is again a P₀-matrix. Simpler conditions are obtained when the rank of the matrix is n?1, where n is the order of the matrix. We then consider the application of these results to singular M-matrices of order n and rank n?1. In particular, for this case we prove that the Moore-Penrose inverse is a P₀-matrix.     

On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements

The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C^∗-algebra elements.     

A note on the Drazin inverses with Banachiewicz-Schur forms

Let H be a Hilbert space, M the closed subspace of H with orthocomplement M^⊥. According to the orthogonal decomposition H=M⊕M^⊥, every operator M∈B(H) can be written in a block-form . In this note, we give necessary and sufficient conditions for a partitioned operator matrix M to have the Drazin inverse with Banachiewicz-Schur form. In addition, this paper investigates the relations among the Drazin inverse, the Moore-Penrose inverse and the group inverse when they can be expressed in the Banachiewicz-Schur forms.     

A “Cramer rule” for the least-norm,least-squared-error solution of inconsistent linear equations

A recent “Cramer rule” for obtaining the least-norm solution of a consistent system of linear equations Ax = b is shown to also provide the least-squared-error solution in the inconsistent case, i.e. the solution A⁺b where A⁺ is the Moore-Penrose inverse of A.     

关于态射的广义Moore-Penrose逆

本文研究了态射的广义Moore-Penrose逆．给出了范畴中态射的广义Moore-Penrose逆存在的一些新的充要条件．也给出了广义Moore-Penrose逆的乘积公式成立的充要条件。     

Generalized inverses of circulant and generalized circulant matrices

An expression for the Moore-Penrose inverse of certain singular circulants by S.R. Searle is generalized to include all circulants. Similar expressions are given for the Moore-Penrose inverse of block circulants with circulant blocks, level-q circulants, k-circulants where |k|=1, and certain other matrices which are the product of a permutation matrix and a circulant. Expressions for other generalized inverses are given.     

Biorthogonal systems based on {2}-inverse with prescribed range and null space

In this paper, the transformations of generating systems of Euclidean space are examined in connection with the 2-inverseA _T,S ⁽²⁾ , S which has prescribed rangeT and null spaceS of their Gram matrices. The biorthogonal systems of the Moore-Penrose inverse and the Drazin inverse are extended to the {2}-inverseA _T,S ⁽²⁾ .     

The Moore-Penrose inverse of a retrocirculant

In a recent paper Chao [2] has determined the eigenvalues of a matrix of the form A=PC where P is a permutation matrix which commutes with a certain unitary matrix and C is a circulant. Here we determine the Moore-Penrose inverse of such a “retrocirculant” and show that the nonzero eigenvalues of the Moore-Penrose inverse are the reciprocals of the nonzero eigenvalues of the retrocirculant.     

Reverse order law for the group inverses

This paper is to present some equivalent conditions concerning the reverse order law #(AB)=B#A# for the group invertible operators A, B on a Hilbert space H.