scholarly journals A condensed Cramer’s rule for the minimum-norm least-squares solution of linear equations

2012 ◽  
Vol 437 (9) ◽  
pp. 2173-2178 ◽  
Author(s):  
Jun Ji
Keyword(s):  
Least Squares ◽  
Linear Equations ◽  
Minimum Norm ◽  
Least Squares Solution ◽  
Cramer’S Rule

scholarly journals A Representation of Nonhomogeneous Quadratic Forms with Application to the Least Squares Solution

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Czesław Stępniak
Keyword(s):  
Least Squares ◽  
Quadratic Forms ◽  
Differential Calculus ◽  
Linear Models ◽  
Linear Equations ◽  
Algebraic Approach ◽  
System Of Linear Equations ◽  
Least Squares Solution ◽  
Inconsistent System ◽  
Usual Solution

The least squares problem appears, among others, in linear models, and it refers to inconsistent system of linear equations. A crucial question is how to reduce the least squares solution in such a system to the usual solution in a consistent one. Traditionally, this is reached by differential calculus. We present a purely algebraic approach to this problem based on some identities for nonhomogeneous quadratic forms.

scholarly journals A novel interpretation of least squares solution

1992 ◽  
Vol 15 (1) ◽  
pp. 41-46
Author(s):  
Jack-Kang Chan
Keyword(s):  
Least Squares ◽  
Source Localization ◽  
Convex Combination ◽  
Linear Equations ◽  
Singular Values ◽  
Statistical Interpretation ◽  
System Of Linear Equations ◽  
Overdetermined System ◽  
The Matrix ◽  
Cramer’S Rule

We show that the well-known least squares (LS) solution of an overdetermined system of linear equations is a convex combination of all the non-trivial solutions weighed by the squares of the corresponding denominator determinants of the Cramer's rule. This Least Squares Decomposition (LSD) gives an alternate statistical interpretation of least squares, as well as another geometric meaning. Furthermore, when the singular values of the matrix of the overdetermined system are not small, the LSD may be able to provide flexible solutions. As an illustration, we apply the LSD to interpret the LS-solution in the problem of source localization.

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