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∈R(Ak)
for A∈Cn×n and b∈Cn, k=Ind(A). When the system is consistent, it is well known that it has a unique ADb. If the system is inconsistent, then we seek for the least squares solution of the problem and consider
min
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.
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