scholarly journals The Group Inverse of the Combinations of Two Idempotent Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Shunqin Wang ◽  
Chunyuan Deng
Keyword(s):  
Group Inverse ◽  
Linear Combinations ◽  
Idempotent Operators ◽  
Group Inverses

We present some inverses and group inverses results for linear combinations of two idempotents and their products.

Weak group inverses in proper ∗-rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2050238 ◽  
Author(s):  
Mengmeng Zhou ◽  
Jianlong Chen ◽  
Yukun Zhou
Keyword(s):  
Group Inverse ◽  
Complex Matrices ◽  
Weak Group ◽  
Group Inverses

In proper ∗-rings, we characterize weak group inverses by three equations. It generalizes the notion of weak group inverse, which was introduced by Wang and Chen for complex matrices in 2018. Some new equivalent characterizations for elements to be weak group invertible are presented. Furthermore, we define the group-EP decomposition. Some properties of the weak group inverse are established by the group-EP decomposition.

A note on the group inverses of block matrices over rings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3685-3692
Author(s):  
Hanyu Zhang
Keyword(s):  
Associative Ring ◽  
Sufficient Conditions ◽  
Block Matrix ◽  
Necessary And Sufficient Conditions ◽  
Group Inverse ◽  
Block Matrices ◽  
Group Inverses ◽  
Necessary And Sufficient

Suppose R is an associative ring with identity 1. The purpose of this paper is to give some necessary and sufficient conditions for the existence and the representations of the group inverse of the block matrix (AX+YB B A 0) and M = (A B C D) under some conditions. Some examples are given to illustrate our results.

The valence bands in two-dimensional graphite

1955 ◽  
Vol 227 (1170) ◽  
pp. 330-349 ◽  
Keyword(s):  
Normal Structure ◽  
Theoretical Treatment ◽  
Two Dimensional ◽  
Linear Combinations ◽  
X Ray ◽  
Atomic Orbitals ◽  
Idempotent Operators ◽  
Valence Bands ◽  
The One

A group-theoretical treatment of the spatial symmetry of two-dimensional graphite leads to a classification of the one-electron eigenstates. The method of idempotent operators is used to derive Bloch orbitals for the crystal valence bands as linear combinations of the atomic orbitals 2 p x , 2 p y and 2 s . The functions 2 p z form the conduction band. The energy of the electrons in such orbitals is estimated in terms of four overlap integrals between nearest neighbour atoms, and four corresponding Hamiltonian integrals. The deduced band structure is not sensitive to the precise values of these integrals, and cannot be changed materially by the inclusion of further neighbours. The states form three touching bands, all fully occupied by electrons in the normal structure. The large band width of some 10 eV affects previous discussions of soft X-ray experiments.

On Group Inverses of Matrices with Order and Rank Perturbations

1994 ◽  
Vol 44 (3-4) ◽  
pp. 209-222 ◽  
Author(s):  
P.S.S.N.V.P. Rao
Keyword(s):  
Markov Chains ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Group Inverse ◽  
Necessary And Sufficient Condition ◽  
Matrix Formula ◽  
Group Inverses ◽  
Finite Markov Chains ◽  
Necessary And Sufficient

Group inverse of a square matrix A exists if and only if rank of A is equal to rank of A2. Group inverses have many applications, prominent among them is in the analysis of finite Markov chains discussed by Meyer (1982). In this note necessary and sufficient conditions for the existence of group inverses of bordered matrix, [Formula: see text] are obtained and expressions for the group inverses in terms of group inverse of A are given, whenever they exist. Also necessary and sufficient condition for the existence of group inverse of A in terms of group inverse of B and C are given. An application to perturbation in Markov chains is illustrated.

Descriptive statistics

2017 ◽  
Author(s):  
Andrew Gelman ◽  
Deborah Nolan
Keyword(s):  
Descriptive Statistics ◽  
Metabolic Rates ◽  
World Record ◽  
Linear Combinations ◽  
Record Times ◽  
Starting Point ◽  
Statistics Course ◽  
Scatter Plots ◽  
Economic Indexes ◽  
The Mean

Descriptive statistics is the typical starting point for a statistics course, and it can be tricky to teach because the material is more difficult than it first appears. The activities in this chapter focus more on the topics of data displays and transformations, rather than the mean, median, and standard deviation, which are covered easily in a textbook and on homework assignments. Specific topics include: distributions and handedness scores; extrapolation of time series and world record times for the mile run; linear combinations and economic indexes; scatter plots and exam scores; and logarithmic transformations and metabolic rates.

scholarly journals Some inequalities of linear combinations of independent random variables: II

Bernoulli ◽  
2013 ◽  
Vol 19 (5A) ◽  
pp. 1776-1789 ◽  
Author(s):  
Xiaoqing Pan ◽  
Maochao Xu ◽  
Taizhong Hu
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Linear Combinations
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