Related papers: The Moore-Penrose inverse in rings with involution
Let $X\in\mathbb{C}^{m\times m}$ and $Y\in\mathbb{C}^{n\times n}$ be nonsingular matrices, and let $N\in\mathbb{C}^{m\times n}$. Explicit expressions for the Moore-Penrose inverses of $M=XNY$ and a two-by-two block matrix, under appropriate…
In the paper we present some new inversion formulas and two new formulas for Stirling numbers.
This paper introduces and studies nil-reversible rings wherein we call a ring R nil-reversible if the left and right annihilators of every nilpotent element of R are equal. Reversible rings (and hence reduced rings) form a proper subclass…
In this paper, we give the first and second fundamental theorems of invariant theory for certain invariant rings whose generators are expressed by circulant determinants.
The aim of this article is to investigate the issues of multiplicative inverses and composition in the set of formal Laurent series. We show the lack of general uniqueness of inverses of formal Laurent series; necessary and sufficient…
The Moore-Penrose algorithm provides a generalized notion of an inverse, applicable to degenerate matrices. In this paper, we introduce a covariant extension of the Moore-Penrose method that permits to deal with general relativity involving…
We propose a proof of the Lagrange Interpolation Formula based on the Chinese Remainder Theorem for arbitrary rings. Even such relationships are known, we think that our viewpoint is worth being published.
We formulate explicitly the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms. Our approach is to apply the method of characteristics of differential equations, by treating…
We determine the rings of invariants in the symmetric algebra on the dual of a vector space V over the field of two elements, for the group G of orthogonal transformations preserving a non-singular quadratic form on V. The invariant ring is…
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…
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…
The Fourier transform is considered as a Henstock--Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The…
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…
A class of inverse problems for a heat equation with involution perturbation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence…
It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that in many cases can be obtained as inverse…
The notion of a Moore-Penrose inverse (M-P inverse) was introduced by Moore in 1920 and rediscovered by Penrose in 1955. The M-P inverse of a complex matrix is a special type of inverse which is unique, always exists, and can be computed…
We present some formulae for the Drazin inverse of difference and product of idempotents in a ring. A number of results of bounded linear operators in Banach spaces are extended to the ring case.
Rotational invariance of physical laws is a generally accepted principle. We show that it leads to an additional external constraint on local realistic models of physical phenomena involving measurements of multiparticle spin 1/2…
Having in mind that physical systems have different levels of structure we develop the concept of external, internal and total improper Lorentz transformation (space inversion and time reversal). A particle obtained from the ordinary one by…
This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.