Year: 2021
Communications in Mathematical Research , Vol. 37 (2021), Iss. 1 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmr.2020-0028
Communications in Mathematical Research , Vol. 37 (2021), Iss. 1 : pp. 86–112
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Bott-Duffin inverse Core-EP inverse weighted core-EP inverse Cramer's rule Gaussian elimination method.