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From: "John D'Errico" <woodchips@rochester.rr.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Matrix inversion problem - What's going on?
Date: Tue, 12 Apr 2011 11:13:05 +0000 (UTC)
Organization: John D'Errico (1-3LEW5R)
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"printer newbie!" <newsboost@gmail.com> wrote in message <1ce4d6d3-488e-4d04-a153-a7137ff652b3@l39g2000yqh.googlegroups.com>...
> On 11 Apr., 18:41, "Florin Neacsu" <fneac...@gmail.com> wrote:
> > "printer newbie!" <newsbo...@gmail.com> wrote in message <82709a89-8b9f-41ca-90b4-c9738d23a...@d11g2000yqo.googlegroups.com>...
> > > Dear all
> >
> > > I've uploaded this pdf-file:
> > >http://www.2shared.com/document/o9q5B5Rw/Matlab_matrix_inversion.html
> >
> > > As you can see, I'm using Matlab:
> >
> > > 1) with two different matrices: A=[1 2; 3 4;5 6] and A=[1 2; 3 4;5 5]
> > > and b=[1;2;3], meaning that:
> > > 2) x=A\b can easily be found
> > > 3) A*x = b, right? NO! Not in the second case!
> >
> > > Any hints/explanations are most welcome. I've put some comments in the
> > > uploaded pdf-file and I guess that those of you who'll answer knows
> > > everything about illposed problems, SVD and possible some other things
> > > that I don't know of. I also tried to find the conditioning number for
> > > the two matrices, however I'm a bit lost and can't come up with a good
> > > explanation myself.
> >
> > As it was said above, this is an overdetermined system. If you are looking for an unique solution, then you system must have one of its lines equal to a linear combination of the two others.
> > In general if you have p equations and n unknowns (with p>n for overdetermined), your system must have n linear independent equations, hence p-n linear dependent eq.
> 
> Ok, thanks. So the number of rows is the number of equations and the
> number of columns is the number of unknowns...
> 
> So it was just a coincidence that it worked out for the first matrix,
> but not the second...
> 
> What property (of the matrix A) is it then, that makes it work out
> great for the first A-matrix, when they're both the same physical size
> (3x2)? Are any lines in the first matrix linear combinations of
> another line?
> 

In the second case, b (the right hand side) was an
exact linear combination of the columns of your first
matrix A. So you might notice that A1(:,2)/2 would
yield [1;2;3]. This just happens to be identically b.

A way to identify that this happens is when

   rank(A) == rank([A,b])

Try it.

A1=[1 2; 3 4;5 6];
A2=[1 2; 3 4;5 5];
b=[1;2;3];

rank(A1) == rank([A1,b])
ans =
     1

rank(A2) == rank([A2,b])
ans =
     0

Therefore the first problem x = A1\b will have an exact
solution, while the second problem does not. Well,
exact to within the floating point precision of MATLAB.

HTH,
John