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Notes on equivalent forms of the general least-squares solution
Abstract The general least-squares solution for a rank-deficient system is expressed in several completely equivalent forms. The simplest form is written in terms of $ A^{+} $, the pseudo-inverse of the design matrix A, and of an arbitrary vector whose presence removes the need for more general inverse operators. One by-product of this development points to the following pitfall: The set of all generalized inverse$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} ^g $$ does not yield all solutions to a consistent system Ax=c by putting$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} ^g $$ c; in particular, the case c=0 is essentially ignored.
Notes on equivalent forms of the general least-squares solution
Abstract The general least-squares solution for a rank-deficient system is expressed in several completely equivalent forms. The simplest form is written in terms of $ A^{+} $, the pseudo-inverse of the design matrix A, and of an arbitrary vector whose presence removes the need for more general inverse operators. One by-product of this development points to the following pitfall: The set of all generalized inverse$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} ^g $$ does not yield all solutions to a consistent system Ax=c by putting$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} ^g $$ c; in particular, the case c=0 is essentially ignored.
Notes on equivalent forms of the general least-squares solution
Blaha, Georges (author)
Bulletin géodésique ; 56
1982
Article (Journal)
English
Geodäsie , Geometrie , Geodynamik , Zeitschrift , Mathematik , Mineralogie
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