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Register a new userThe Moore-Penrose inverses of split quaternions
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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.
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Unlike the Hamilton quaternion algebra, the split-quaternions contain nontrivial zero divisors. In general speaking, it is hard to find the solutions of equations in algebras containing zero divisor. In this paper, we manage to derive explicit formulas for computing the roots of $x^{2}+bx+c=0$ in split quaternion algebra.
For two positive definite adjointable operators $M$ and $N$, and an adjointable operator $A$ acting on a Hilbert $C^*$-module, some properties of the weighted Moore-Penrose inverse $A^dag_{MN}$ are established. When $A=(A_{ij})$ is $1times 2$ or $2times 2$ partitioned, general representations for $A^dag_{MN}$ in terms of the individual blocks $A_{ij}$ are studied. In case $A$ is $1times 2$ partitioned, a unified representation for $A^dag_{MN}$ is presented. In the $2times 2$ partitioned case, an approach to constructing Moore-Penrose inverses from the non-weighted case to the weighted case is provided. Some results known for matrices are generalized in the general setting of Hilbert $C^*$-module operators.
In this paper, we introduce the notion of the Hom-Leibniz-Rinehart algebra as an algebraic analogue of Hom-Leibniz algebroid, and prove that such an arbitrary split regular Hom-Leibniz-Rinehart algebra $L$ is of the form $L=U+sum_gamma I_gamma$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_gamma$, a well described ideal of $L$, satisfying $[I_gamma, I_delta]= 0$ if $[gamma] eq [delta]$. In the sequel, we develop techniques of connections of roots and weights for split Hom-Leibniz-Rinehart algebras respectively. Finally, we study the structures of tight split regular Hom-Leibniz-Rinehart algebras.
In this paper, the problems of perturbation and expression for the Moore--Penrose metric generalized inverses of bounded linear operators on Banach spaces are further studied. By means of certain geometric assumptions of Banach spaces, we first give some equivalent conditions for the Moore--Penrose metric generalized inverse of perturbed operator to have the simplest expression $T^M(I+ delta TT^M)^{-1}$. Then, as an application our results, we investigate the stability of some operator equations in Banach spaces under different type perturbations.
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.
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