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 lumen8r February 3rd, 2010 01:13 PM

Linear equation problem

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This doesn't seem too hard, but I just can't figure it out... any help would be greatly appreciated! Here it goes:

I have a matrix $M$ of measured data points ($m \times n\$, i.e. $m$ rows and $n$ comlumns).
I know that the matrix is made up of the product of a column vector $\overline{a}$ with a row vector $\overline{x}$, added to the product of a column vector of ones with another row vector

$\overline{y}$. In other words, $M_{kl}=a_kx_l+y_l$ and the $a_k$, $k=1\ldots m$ are the same within each row and the $x_l$ and $y_l$, $l=1\ldots n$ are the same within each column.

The $M_{kl}$ are known (measured) and the $a_k$, $x_l$ and $y_l$ are unknown. Therefore there are $m\cdot n$ equations and $m+2n$ unknown values. For example a $3\times 3$ matrix would provide 9 equations for 9 variables and there should be an exact solution. Any larger matrix would be overdetermined and should provide redundant equations for the $a_k$, $x_l$ and $y_l$ (which could be used for calculating averages and estimating measurement errors).

How can the $a_k$, $x_l$ and $y_l$ be calculated? Thanks again for any help or advice on this!

(for some reason the latex \overrightarrow did not work, so I used \overline to designate a vector...)

 lumen8r March 9th, 2010 03:06 PM

Re: Linear equation problem

Hi Everyone - no replies yet? Maybe the problem is harder than I thought...?

Any thoughts or ideas would be very much appreciated, though: Maybe it was just a dumb question to ask? Maybe this is not solvable because of some fundamental/logical reasons? Or, if this is not obviously the case, maybe you know of another place where I could post this question?

Thanks again!

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