Matrix decomposition again If some matrix (M×N) can be expressed as product of (M×1) and (1×N) vectors:



*

*what is proper term for such kind of decomposition?

*how to tell if such kind of decomposition exists for given matrix?

*how to find values of these vectors using numerical methods?

 A: There are several related terms for this type of factorization and it will probably depend on context as to which one is the most appropriate.
First, you can call it the Kronecker product of the two matrices and more specifically you can call this an outer product.
You can also say that this is a rank factorization of the matrix: Any $m\times n$ matrix with rank $r$ can be factored into a product of a $m\times r$ matrix with a $r \times n$ matrix. As Gerry has mentioned, your matrix is the special case of a rank $1$ matrix being factored into column and row vectors.
A: It only exists if the matrix is of rank one, so if every row is a multiple of any (nonzero) row. So pick a now zero row and see whether every other row is a multiple of it. 
A: These are called dyadics or matrices of rank $1$.
Every matrix can be written as the sum of dyadic matrices, and starting from the first nonzero culomn and row you can separate down the first summand, and this procedure also gives the rank of the matrix (as the number of summands), if done optimally.
