TITLE: Is it possible to decompose a matrix as the product of two vectors?
QUESTION [10 upvotes]: This question maybe stupid to some of you, but I would like to know whether it is possible to decompose a matrix $M_{m\times n}$ as the product of two vectors, i.e.
$$M_{m\times n} = \vec{y}_{m\times 1}\times\vec{x}_{1\times n}+const.$$ 
Obviously, this should be true for some cases, but I am not sure whether this conclusion is always hold. Meanwhile, I want to know at what condition, we can do this decomposition and how to find the vectors $\vec{y}$ and $\vec{x}$?

REPLY [6 votes]: In general, rank $n$ matrix can be expressed as  sum of $n$ rank 1 matrix using singular value decomposition. 
When the matrix is rank 1 we can express as you suggested in your question.
If it is $>1$, we can't express like that.