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Volume 37, Issue 1
The Core-EP, Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations

Jun Ji & Yimin Wei

Commun. Math. Res., 37 (2021), pp. 86-112.

Published online: 2021-02

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  • Abstract

We study the constrained system of linear equations

$Ax=b$, $x∈\mathcal{R}(A^k)$

for  $A ∈ \mathbb{C}^{n×n}$  and  $b ∈\mathbb{C}^n$, $k=Ind(A)$. When the system is consistent, it is well known that it has a unique $A^Db$. If the system is inconsistent, then we seek for the least squares solution of the problem and consider
$$\min _{x \in \mathcal{R}\left(A^{k}\right)}\|b-A x\|{_2,}$$

where $\|\cdot \|_2$ is  the 2-norm. For the inconsistent system with a matrix $A$ of index one, it was proved recently that the solution is $A^⊕b$ using the core inverse $A^⊕$ of $A$. For matrices of an arbitrary index and an arbitrary $b$, we show that the solution of the constrained system can be expressed as $A^⊕b$ where $A^⊕$ is the core-EP inverse of $A$. We establish two Cramer's rules for the inconsistent constrained least squares solution and develop several explicit expressions  for the core-EP inverse of matrices of an arbitrary index. Using these expressions, two Cramer's rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper. We also consider the $W$-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.


  • AMS Subject Headings

15A09, 15A24, 15A29, 15A57

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jji@kennesaw.e (Jun Ji)

  • BibTex
  • RIS
  • TXT
@Article{CMR-37-86, author = {Ji , Jun and Wei , Yimin}, title = {The Core-EP, Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {37}, number = {1}, pages = {86--112}, abstract = {

We study the constrained system of linear equations

$Ax=b$, $x∈\mathcal{R}(A^k)$

for  $A ∈ \mathbb{C}^{n×n}$  and  $b ∈\mathbb{C}^n$, $k=Ind(A)$. When the system is consistent, it is well known that it has a unique $A^Db$. If the system is inconsistent, then we seek for the least squares solution of the problem and consider
$$\min _{x \in \mathcal{R}\left(A^{k}\right)}\|b-A x\|{_2,}$$

where $\|\cdot \|_2$ is  the 2-norm. For the inconsistent system with a matrix $A$ of index one, it was proved recently that the solution is $A^⊕b$ using the core inverse $A^⊕$ of $A$. For matrices of an arbitrary index and an arbitrary $b$, we show that the solution of the constrained system can be expressed as $A^⊕b$ where $A^⊕$ is the core-EP inverse of $A$. We establish two Cramer's rules for the inconsistent constrained least squares solution and develop several explicit expressions  for the core-EP inverse of matrices of an arbitrary index. Using these expressions, two Cramer's rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper. We also consider the $W$-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.


}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0028}, url = {http://global-sci.org/intro/article_detail/cmr/18626.html} }
TY - JOUR T1 - The Core-EP, Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations AU - Ji , Jun AU - Wei , Yimin JO - Communications in Mathematical Research VL - 1 SP - 86 EP - 112 PY - 2021 DA - 2021/02 SN - 37 DO - http://doi.org/10.4208/cmr.2020-0028 UR - https://global-sci.org/intro/article_detail/cmr/18626.html KW - Bott-Duffin inverse, Core-EP inverse, weighted core-EP inverse, Cramer's rule, Gaussian elimination method. AB -

We study the constrained system of linear equations

$Ax=b$, $x∈\mathcal{R}(A^k)$

for  $A ∈ \mathbb{C}^{n×n}$  and  $b ∈\mathbb{C}^n$, $k=Ind(A)$. When the system is consistent, it is well known that it has a unique $A^Db$. If the system is inconsistent, then we seek for the least squares solution of the problem and consider
$$\min _{x \in \mathcal{R}\left(A^{k}\right)}\|b-A x\|{_2,}$$

where $\|\cdot \|_2$ is  the 2-norm. For the inconsistent system with a matrix $A$ of index one, it was proved recently that the solution is $A^⊕b$ using the core inverse $A^⊕$ of $A$. For matrices of an arbitrary index and an arbitrary $b$, we show that the solution of the constrained system can be expressed as $A^⊕b$ where $A^⊕$ is the core-EP inverse of $A$. We establish two Cramer's rules for the inconsistent constrained least squares solution and develop several explicit expressions  for the core-EP inverse of matrices of an arbitrary index. Using these expressions, two Cramer's rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper. We also consider the $W$-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.


JunJi & YiminWei. (2021). The Core-EP, Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations. Communications in Mathematical Research . 37 (1). 86-112. doi:10.4208/cmr.2020-0028
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