Re: Solving matrix equation

• To: mathgroup at smc.vnet.net
• Subject: [mg17097] Re: [mg17176] Solving matrix equation
• From: "Mark E. Harder" <harderm at ucs.orst.edu>
• Date: Fri, 23 Apr 1999 02:32:24 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```Oldfield,
I think it would help us if you provided more information.  Are you
trying to solve for A, given a fixed M? or the other way round?  Is the
problem numeric or symbolic (exact)?  What are the dimensions of M?
I'll assume for now that you are solving for A.  You say that M is
rectangular, which usually means that the # of rows does not = # of columns.
But then we need to ask: How does the matrix M which could have different
row and column dimensions represent a linear operator which maps the vector
space which contains A unto itself?  Usually, such an operator would be
represented by a SQUARE matrix, and then your problem would be an
eigenvalue-eigenvector problem, i.e. the solution for A is the set of
eigenvectors for M which have 1 as their eigenvalue.
To multiply A from the left, the number of columns in M must be the same
as the # of rows in A, call it n.  Then if the # of rows,m, in M is not =n,
& either m>n or m<n. In the former case, MA=A represents an overdetermined
system, which will have no exact solution, unless exactly n out of m rows
are linearly independent.  The latter case (m<n) represents an
underdetermined system, which has an infinite solution set.
I believe that one of these problems is the cause of your error message.
If you are looking for an approximate numerical solution, then there are
tools in Mtca. for dealing with your problem, but I am hesitant to say more
without knowing more about your problem.  I also am a little shaky in my
knowledge of Linear Algebra, and it might be better if someone more
knowledgeable helps you  -- once you explain things a little clearer.
Good Luck,
mark harder

-----Original Message-----
From: Oldfield So <kysoNOspam at cse.cuhk.eduDOThk>
To: mathgroup at smc.vnet.net
Subject: [mg17097] [mg17176] Solving matrix equation

>Dear all,
>
>I need to solve a equation in the following format:
>
> M A = A
>
>where M is a rectangular matrix, A is a column vector.  By using Solve[],
>the following "error" is obtained:
>
>RowReduce::luc:
>   Result for RowReduce of badly conditioned matrix {<<12>>}
>     may contain significant numerical errors.
>
>I would like to know is there any standard way to solve it?  Thanks a lot.
>
>Best Regards,
>Oldfield So
>--
>Oldfield K.Y. SO | kysoNOspam at cse.cuhk.eduDOThk |
http://www.cse.cuhk.edu.hk/~kyso
>

```

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