$2\times2$ block representations of the Moore-Penrose inverse and orthogonal projection matrices

The Moore-Penrose inverse of differences and products of projectors in a ring with involution

In this paper, we study the Moore-Penrose inverses of differences and products of projectors in a ring with involution. Also, some necessary and sufficient conditions for the existence of such inverses are given, and their expressions are presented.

张量的加权Moore--Penrose逆的反序律

Reverse-order law for weighted Moore--Penrose inverse of tensors

本文研究任意阶张量加权Moore--Penrose逆的性质，给出偶阶方张量反序律的新充分条件，并刻画任意阶张量的反序律。

The Moore-Penrose Inverses of Clifford Algebra $C\ell_{1,2}$

In this paper, we introduce a ring isomorphism between the Clifford algebra $C\ell_{1,2}$ and a ring of matrices. By such a ring isomorphism, we introduce the concept of the Moore-Penrose inverse in Clifford algebra $C\ell_{1,2}$. Using the Moore-Penrose inverse, we solve the linear equation $axb=d$ in $C\ell_{1,2}$. We also obtain necessary and sufficient conditions for two numbers in $C\ell_{1,2}$ to be similar.

Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer's rules

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 analogs of the adjoint matrix, we get the Cramer rules for the least squares solution of left and right systems of quaternionic linear equations.

The Moore-Penrose Inverses of Clifford Algebra $C\ell_2$

In this paper, we introduce a ring isomorphism between the Clifford algebra $C\ell_2$ and a ring of matrices, and represent the elements in $C\ell_2$ by real matrices. By such a ring isomorphism, we introduce the concept of the Moore-Penrose inverse in Clifford algebra $C\ell_2$. we solve the linear equation $axb=d$, $ax=xb$ and $ax=\bar{x}b$. We also obtain necessary and sufficient conditions for two numbers in $C\ell_2$ to be similar and pseudosimilar.

Moore-Penrose Invertibility of Differences and Products of Projections in Rings with Involution

This article concerns the MP inverse of the differences and the products of projections in a ring $R$ with involution. Some equivalent conditions are obtained. As applications, the MP invertibility of the commutator $pq-qp$ and the anti-commutator $pq+qp$ are characterized, where $p$ and $q$ are projections in $R$. Some related known results in $C^*$-algebra are generalized.

Projections and Idempotents in star-reducing Rings Involving the Moore-Penrose Inverse

In [Comput. Math. Appl. 59 (2010) 764-778], Baksalary and Trenkler characterized some complex idempotent matrices of the form $(I-PQ)^†P(I-Q)$, $(I-QP)^†(I-Q)$ and $P(P+Q-QP)^†$ in terms of the column spaces and null spaces of $P$ and $Q$, where $P, Q\in \mathbb{C}^{\textsf{OP}}_n = \{L\in \mathbb{C}_{n,n}\ |\ L^2 = L = L^*\}$. We generalize these results from $\mathbb{C}_{n,n}$ to any *-reducing rings.

Determinantal representations of the quaternion weighted Moore-Penrose inverse and corresponding Cramer's rule

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 noncommutative column-row determinants. By using the obtained analogs of the adjoint matrix, we get the Cramer rules for the weighted Moore-Penrose solutions of left and right systems of quaternion linear equations.

关于张量的反序律及其在Moore-Penrose逆加法结果中的应用

On reverse-order law of tensors and its application to additive results on Moore-Penrose inverse

该研究重新探讨基于Einstein乘积的张量Moore-Penrose逆反序律，给出任意阶张量反序律的新刻画，将其应用于张量和的Moore-Penrose逆，推导Frobenius范数下的加性扰动界，并提出张量亚真分裂概念以辅助张量多线性系统迭代求解。

具有treeangle性质的矩阵的Moore-Penrose逆的图

Graphs of Moore-Penrose inverse of matrices possessing the treeangle property

本文研究具有treeangle性质的矩阵的Moore-Penrose逆的图结构，证明其与可逆情形不同，并给出精确刻画。

Moore-Penrose inverse, parabolic subgroups, and Jordan pairs

A Moore--Penrose inverse of an arbitrary complex matrix A is defined as a unique matrix A' such that AA'A=A, A'AA'=A', and AA', A'A are Hermite matrices. We show that this definition has a natural generalization in the context of shortly graded simple Lie algebras corresponding to parabolic subgroups with aura (abelian unipotent radical) in simple complex Lie groups, or equivalently in the context of simple complex Jordan pairs. We give further generalizations and applications.

The Moore-Penrose inverses of split quaternions

In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations $axb=d$, $xa=bx$ and $xa=b\bar{x}$. Also we obtain necessary and sufficient conditions for two split quaternions to be similar or consimilar.

Group, Moore-Penrose, core and dual core inverse in rings with involution

Let $R$ be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary $*$-ring case. It is shown that the group, Moore-Penrose, core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in $R$ are obtained. It is shown that core and dual core inverse belong to the class of inverses along…

The generalized inverses of the quaternion tensor via the T-product

In this article, specific definitions of the Moore-Penrose inverse, Drazin inverse of the quaternion tensor and the inverse along two quaternion tensors are introduced under the T-product. Some characterizations, representations and properties of the defined inverses are investigated. Moreover, algorithms are established for computing the Moore-Penrose inverse, Drazin inverse of the quaternion tensor and the inverse along two quaternion tensors, respectively.

Representations and properties of the W-weighted core-EP inverse

In this paper, we investigate the weighted core-EP inverse introduced by Ferreyra, Levis and Thome. Several computational representations of the weighted core-EP inverse are obtained in terms of singular-value decomposition, full-rank decomposition and QR decomposition. These representations are expressed in terms of various matrix powers as well as matrix product involving the core-EP inverse, Moore-Penrose inverse and usual matrix inverse. Finally, those representations involving only Moore-Penrose inverse are compared and analyzed via computational complexity and numerical examples.

Further results on generalized inverses of tensors via the Einstein product

The notion of the Moore-Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore-Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.

EP elements in rings with involution

Let $R$ be a unital ring with involution. We first show that the EP elements in $R$ can be characterized by three equations. Namely, let $a\in R$, then $a$ is EP if and only if there exists $x\in R$ such that $(xa)^{\ast}=xa$, $xa^{2}=a$ and $ax^{2}=x.$ It is well known that all EP elements in $R$ are core invertible and Moore-Penrose invertible. We give more equivalent conditions for a core (Moore-Penrose) invertible element to be an EP element. Finally, the EP elements are characterized in terms of $n$-EP property, which is a generalization of bi-EP property.

通过特殊化在有理函数域上有效计算 Moore-Penrose 逆

Effective computation of Moore-Penrose inverses over fields of rational functions by specializations

本文针对有理函数域上的矩阵，通过特殊化方法，仅用线性代数求解 Penrose 条件，并精确刻画伪逆失效的参数值（即秩下降点），给出符号算法并验证于经济模型。

基于C-乘积的张量广义逆

The generalized inverses of tensors via the C-Product

该文针对C-乘积下的张量广义逆问题，定义了三类张量广义逆，推导了其基于张量分解的多种表达式，并建立了对应的求解算法。

Rank Equalities Related to Generalized Inverses of Matrices and Their Applications

This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions …

Banach代数中的广义加权EP元素

Generalized weighted EP elements in Banach algebras

本文提出一类新的广义加权逆，作为Banach *-代数中EP和*-DMP元素的推广，研究了其特征、core-EP分解及类极化性质，并揭示了*-DMP元素的新性质。

关于与某些图类相关的矩阵的广义逆

On generalized inverses of matrices associated with certain graph classes

本文研究双星有向图与D-连接星有向图相关矩阵的广义逆，给出了Drazin逆、群逆和Moore-Penrose逆的存在条件及显式公式，揭示了代数结构与图类组合性质的联系。

Cramer's Rule for Generalized Inverse Solutions of Some Matrices Equations

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 the Moore Penrose, weighted Moore-Penrose, Drazin and weighted Drazin inverses. New determinantal representations of these generalized inverse based on their limit representations are introduced in this paper. Application of this new method allows us…

Characterizations and representations of core and dual core inverses

In this paper, double commutativity and the reverse order law for the core inverse are considered. Then, new characterizations of the Moore-Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

New determinantal representations of the W-weighted Drazin inverse over the quaternion skew field

Within the framework of the theory of the column and row determinants, we obtain new determinantal representations of the W-weighted Drazin inverse over the quaternion skew field. We give determinantal representations of the W-weighted Drazin inverse by using previously introduced determinantal representations of the Drazin inverse, the Moore-Penrose inverse, and the limit representations of the W-weighted Drazin inverse in some special case.

A new generalized inverse of matrices from core-EP decomposition

A new generalized inverse for a square matrix $H\in\mathbb{C}^{n\times n}$, called CCE-inverse, is established by the core-EP decomposition and Moore-Penrose inverse $H^†$. We propose some characterizations of the CCE-inverse. Furthermore, two canonical forms of the CCE-inverse are presented. At last, we introduce the definitions of CCE-matrices and $k$-CCE matrices, and prove that CCE-matrices are the same as $i$-EP matrices studied by Wang and Liu in [The weak group matrix, Aequationes Mathematicae, 93(6): 1261-1273, 2019].

The quaternion core inverse and its generalizations

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 necessary tools to represent these generalized inverses, we use their determinantal representations previously obtained by using row-column determinants. As the special case, we give their determinantal representations for matrices with complex entries …

Block representations of generalized inverses of matrices

In this paper will be considered standard forms of generalized inverses for matrices in the shape of block representations {1, 2, 3, 4, 5, 5^k}-inverse. Especially will be considered Moore-Penrose inverse and the group inverse. Results from Rhode's technique are used and methods for calculating some inverse are shown on examples.

核心EP、双核EP和一类结构矩阵的复合广义反演

Core EP, Dual Core EP and Composite Generalized Inverses for a Class of Structured Matrices

研究双星有向图关联矩阵的广义反演，推导核心、双核、核心EP和双核EP的显式块公式及存在条件，结合Moore-Penrose和核心型反演得到其他复合伪反演。