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I have a m*n matrix which i know is shaped like this :
if a1, a2, ..., am, and b1, b2, ..., bn are real numbers, then for any matrix coefficient m(i,j) = ai*bj

Preliminary question : how do you call such product of two vectors that builds a matrix ? Thx to let me know so i'll update the question.

But, besides naming, here's the issue :
1) i get the m(i,j) from a measure, so there's a noise added. (less than 2% if that matters)
2) some measures are missing <=> the matrix is sparse. Still i have at least one measure for each line/column.

Question : How could i get the ai and bj coefficients for this matrix ?

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A matrix $M$ with $m(i,j)=a_i b_j$ is called the outer product of vectors $a$ and $b$, often denoted $a\otimes b$. A matrix has this form if and only if its rank is at most $1$. Thus, your problem can be stated as: fit a rank-1 matrix to given (incomplete) data.

When all entries of the matrix are known, this is a special case of low-rank approximation, solved with SVD.

With incomplete data, this is low-rank matrix recovery (LRMR), a subject of current interest in compressed sensing, data mining and whatnot. You will find several methods, including code, at the website Low-Rank Matrix Recovery and Completion via Convex Optimization maintained by Yi Ma group at Illinois.

Also, this question would be better placed at SciComp.SE. Here is a relevant question from there: Exact recovery of large incomplete rank-one matrices?

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