In this paper,we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse:\begin{align}\nonumber \left\|{Mx - b} \right\|_F=\min\ \ {\rm subject\ to} \ \ {x\in\mathcal{R}(M)} ,\end{align} where $M\in\mathbb{C}^{\texttt{CM}}_n$. We get the unique solution to the problem, provide two Cramer's rules for the unique solution, and establish two new expressions for the core inverse.
In this paper, we introduce the notion of a (generalized) right core inverse and give its characterizations and expressions. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses and EP elements.
Since the day the core inverse has been known in a paper of Bakasarly and Trenkler, it has been widely researched. So far, there are four generalizations of this inverse for the case of matrices of an arbitrary index, namely, the BT inverse, the DMP inverse, the core-EP inverse and the WC inverse. In this paper we introduce a new type of generalized inverse for a matrix of arbitrary index to be called $m$-weak core inverse which generalizes the core-EP inverse, the WC inverse, and therefore the core inverse. We study several properties and characterizations of the $m$-weak core inverse by usin…
In this paper, we introduce two new generalized inverses of matrices, namely, the $\bra{i}{m}$-core inverse and the $\pare{j}{m}$-core inverse. The $\bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler \cite{BT} and the core-EP inverse defined by Manjunatha Prasad and Mohana \cite{MM}. The $\pare{j}{m}$-core inverse of a complex matrix extends the notions of the core inverse and the ${\rm DMP}$-inverse defined by Malik and Thome \cite{MT}. Moreover, the formulae and properties of these two new concepts are investigated by using …
Let $R$ be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in $R$. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary $*$-ring case by Rakić, Dinčić and Djordjević in 2014. Our definition of pesudo core inverse extends the notion of core inverse to elements of an arbitrary index in $R$. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of $*$-ring. Some eq…
In this paper, we present three limit representations of the core-EP inverse. The first approach is based on the full-rank decomposition of a given matrix. The second and third approaches, which depend on the explicit expression of the core-EP inverse, are established. The corresponding limit representations of the dual core-EP inverse are also given. In particular, limit representations of the core and dual core inverse are derived
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.
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.
In this paper, firstly we study the continuity of the core-EP inverse without explicit error bounds by virtue of two methods. One is the rank equality, followed from the classical generalized inverse. The other one is matrix decomposition. The continuity of the core inverse can be derived as a particular case. Secondly, we study perturbation bounds for the core-EP inverse under prescribed conditions. Perturbation bounds for the core inverse can be derived as a particular case. Also, as corollaries, the sufficient (and necessary) conditions for the continuity of the core-EP inverse are obtained…
Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $φ: X \rightarrow X$ is a morphism with kernel $κ: K \rightarrow X$ in $\mathscr{C}$, then $φ$ is core invertible if and only if $φ$ has a cokernel $λ: X \rightarrow L$ and both $κλ$ and $φ^{\ast}φ^3+κ^{\ast}κ$ are invertible. In this case, we give the representation of the core inverse of $φ$. We also give the corresponding result about dual core inverse.
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 …
The main purpose of this paper is to provide a solution of the consistent fuzzy linear system and a generalized solution of the inconsistent fuzzy linear system involving the core-EP inverse of an associated matrix. Before this can be achieved, it is necessary to study the block structure of the core-EP inverse. Finally, results are illustrated with some numerical examples.
The notion of the weighted core inverse in a ring with involution was introduced, recently [Mosic et al. Comm. Algebra, 2018; 46(6); 2332-2345]. In this paper, we explore new representation and characterization of the weighted core inverse of sum and difference of two weighted core invertible elements in an unital ring with involution under different conditions. Further, we discuss reverse order laws and mixed-type reverse order laws for the weighted core invertible elements in a ring.
The core inverse for a complex matrix was introduced by Baksalary and Trenkler. Rakić, Dinčić and Djordjević generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible, in this paper, we will answer this question. We will use three equations to characterize the core inverse of an element. That is, let $a, b\in R$, then $a\in R^{\tiny\textcirc…
Let $\mathscr{C}$ be a category with an involution $\ast$. Suppose that $φ: X \rightarrow X$ is a morphism and $(φ_1, Z, φ_2)$ is an (epic, monic) factorization of $φ$ through $Z$, then $φ$ is core invertible if and only if $(φ^{\ast})^2φ_1$ and $φ_2φ_1$ are both left invertible if and only if $((φ^{\ast})^2φ_1, Z, φ_2)$, $(φ_2^{\ast}, Z, φ_1^{\ast}φ^{\ast}φ)$ and $(φ^{\ast}φ_2^{\ast}, Z, φ_1^{\ast}φ)$ are all essentially unique (epic, monic) factorizations of $(φ^{\ast})^2φ$ through $Z$. We also give the corresponding result about dual core inverse. In addition, we give some characterizations…
We present new additive results for the core inverse in a Banach algebra with involution. We obtain necessary and sufficient conditions under which the sum of two core invertible elements in a Banach algebra with involution is core invertible. Then we apply our results to block complex matrices and obtain certain conditions under which a block complex is core invertible. These generalize many known results, e.g.,~\cite[Theorem 4.3]{XCZ}, ~\cite[Theorem 2.5]{XS}.
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].
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 representations by using noncommutative column and row determinants previously introduced by the author. As the special cases, by using the usual determinant, we give their determinantal representations for matrices with complex entries as well. A numerical examp…
In this paper, we introduce the dual index and dual core generalized inverse (DCGI). By applying rank equation, generalized inverse and matrix decomposition, we give several characterizations of the dual index when it is equal to one. And we get that if DCGI exists, then it is unique. We derive a compact formula for DCGI and a series of equivalent characterizations of the existence of the inverse. It is worth nothing that the dual index of $\hat{M}$ is equal to one if and only if its DCGI exists. When the dual index of $\hat{M}$ is equal to one, we study dual Moore-Penrose generalized inverse …
Let R be a unital ring with involution, we give the characterizations and representations of the core and dual core inverses of an element in R by Hermitian elements (or projections) and units. For example, let a in R and n is an integer greater than or equal to 1, then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that pa=0, a^n+p is invertible. As a consequence, a is an EP element if and only if there exists a Hermitian element (or a projection) p such that pa=ap=0, a^n+p is invertible. We also get a new characterization for both core invertibl…
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…
In this paper, we introduce the m-core-EP inverse in Minkowski space, consider its properties, and get several sufficient and necessary conditions for the existence of the m-core-EP inverse. We give the m-core-EP decomposition in Minkowski space, and note that not every square matrix has the decomposition. Furthermore, by applying the m-core-EP inverse and the m-core-EP decomposition, we introduce the m-core-EP order and give some characterizations of it.
$R$ is a unital ring with involution. We investigate the characterizations and representations of weighted core inverse of an element in $R$ by idempotents and units. For example, let $a\in R$ and $e\in R$ be an invertible Hermitian element, $n\geqslant 1$, then $a$ is $e$-core invertible if and only if there exists an element (or an idempotent) $p$ such that $(ep)^{\ast}=ep$, $pa=0$ and $a^{n}+p$ (or $a^{n}(1-p)+p$) is invertible. As a consequence, let $e, f\in R$ be two invertible Hermitian elements, then $a$ is weighted-$\mathrm{EP}$ with respect to $(e, f)$ if and only if there exists an e…
Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $φ: X \rightarrow X$ is a morphism of $\mathscr{C}$ with core inverse $φ^{\co} : X \rightarrow X$ and $η: X \rightarrow X$ is a morphism of $\mathscr{C}$ such that $1_X+φ^{\co}η$ is invertible. Let $α=(1_X+φ^{\co}η)^{-1},$ $β=(1_X+ηφ^{\co})^{-1},$ $\varepsilon=(1_X-φφ^{\co})ηα(1_X-φ^{\co}φ),$ $γ=α(1_X-φ^{\co}φ)β^{-1}φφ^{\co}β,$ $σ=αφ^{\co}φα^{-1}(1_X-φφ^{\co})β,$ $δ=β^{\ast}(φ^{\co})^{\ast}η^{\ast}(1_X-φφ^{\co})β.$ Then $f=φ+η-\varepsilon$ has a core inverse if and only if $1_X-γ$, $1_X-σ$ and $1_X-δ$ are inverti…
In this paper, we investigate *-DMP elements in $*$-semigroups and $*$-rings. The notion of *-DMP element was introduced by Patrício in 2004. An element $a$ is *-DMP if there exists a positive integer $m$ such that $a^{m}$ is EP. We first characterize *-DMP elements in terms of the \{1,3\}-inverse, Drazin inverse and pseudo core inverse, respectively. Then, we give the pseudo core decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for matrices to an arbitrary $*$-ring; and this decomposition turns to be a useful tool to characterize *-DM…
Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse in a semigroup, but also the Koliha-Drazin inverse in a ring. In this setting we get a core decomposition similar to the nilpotent, Kato or Mbekhta decompositions. In Banach and Operator algebras, we show that the study of the spectrum is not sufficient, and use ideas from local spectral theory to study this new inverse.
