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Register a new userOn the existence of group inverses of Peirce corner matrices
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We give some statements that are equivalent to the existence of group inverses of Peirce corner matrices of a $2 times 2$ block matrix and its generalized Schur complements. As applications, several new results for the Drazin inverses of the generalized Schur complements and the $2 times 2$ block matrix are obtained and some of them generalize several results in the literature.
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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.
Let $R$ be a Bezout domain, and let $A,B,Cin R^{ntimes n}$ with $ABA=ACA$. If $AB$ and $CA$ are group invertible, we prove that $AB$ is similar to $CA$. Moreover, we have $(AB)^{#}$ is similar to $(CA)^{#}$. This generalize the main result of Cao and Li(Group inverses for matrices over a Bezout domain, {it Electronic J. Linear Algebra}, {bf 18}(2009), 600--612).
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 $xa=bbar{x}$. Also we obtain necessary and sufficient conditions for two split quaternions to be similar or consimilar.
Let $mathscr{C}$ be an additive category with an involution $ast$. Suppose that $varphi : X rightarrow X$ is a morphism with kernel $kappa : K rightarrow X$ in $mathscr{C}$, then $varphi$ is core invertible if and only if $varphi$ has a cokernel $lambda: X rightarrow L$ and both $kappalambda$ and $varphi^{ast}varphi^3+kappa^{ast}kappa$ are invertible. In this case, we give the representation of the core inverse of $varphi$. We also give the corresponding result about dual 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.
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