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Register a new userA new generalized inverse of matrices from core-EP decomposition
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A new generalized inverse for a square matrix $Hinmathbb{C}^{ntimes n}$, called CCE-inverse, is established by the core-EP decomposition and Moore-Penrose inverse $H^{dag}$. 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].
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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, 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 matrix decompositions and matrix powers.
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.
A matrix $P$ is said to be a nontrivial generalized reflection matrix over the real quaternion algebra $mathbb{H}$ if $P^{ast }=P eq I$ and $P^{2}=I$ where $ast$ means conjugate and transpose. We say that $Ainmathbb{H}^{ntimes n}$ is generalized reflexive (or generalized antireflexive) with respect to the matrix pair $(P,Q)$ if $A=PAQ$ $($or $A=-PAQ)$ where $P$ and $Q$ are two nontrivial generalized reflection matrices of demension $n$. Let ${large varphi}$ be one of the following subsets of $mathbb{H}^{ntimes n}$ : (i) generalized reflexive matrix; (ii)reflexive matrix; (iii) generalized antireflexive matrix; (iiii) antireflexive matrix. Let $Zinmathbb{H}^{ntimes m}$ with rank$left( Zright) =m$ and $Lambda=$ diag$left( lambda_{1},...,lambda_{m}right) .$ The inverse eigenproblem is to find a matrix $A$ such that the set ${large varphi }left( Z,Lambdaright) =left{ Ain{large varphi}text{ }|text{ }AZ=ZLambdaright} $ nonempty and find the general expression of $A.$ ewline In this paper, we investigate the inverse eigenproblem ${large varphi}left( Z,Lambdaright) $. Moreover, the approximation problem: $underset{Ain{large varphi}}{minleftVert A-ErightVert _{F}}$ is studied, where $E$ is a given matrix over $mathbb{H}$ and $parallel cdotparallel_{F}$ is the Frobenius norm.
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.
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