Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\vec{x}$?

share|cite|improve this question
You should look into the singular value decomposition, which is essentially a way to write a rank-$k$ matrix as a sum of $k$ such terms, $\mathbf M=\sum\limits_{i=1}^k\sigma_i\mathbf u_i\mathbf v_i^T$. In particular, you can write $\mathbf M$ as a single term $\sigma\mathbf u\mathbf v^T$ if and only if $\mathbf M$ is rank $1$. (If you don't like the presence of the extra scalar $\sigma$, you can think of $\sigma\mathbf u$ and $\mathbf v$, or $\mathbf u$ and $\sigma\mathbf v$, as your two vectors.) – Rahul Mar 10 '13 at 2:02
Much better than either of the proposed answers, IMO. – bubba Mar 10 '13 at 2:31
@RahulNarain: Well, or rank zero. :-) – cardinal Mar 10 '13 at 3:31
@TimSeguine That question is specific to symmetric matrices, and the answer there uses that assumption. Here we have a rectangular matrix. – user147263 May 25 '14 at 22:26

If you by $\times$ mean the cross product, then this of course doesn't make sense.

If you mean a matrix product, then this also will not work. Take for example $$ \pmatrix{1 & 0 \\ 0 & 1} $$ and assume that $$ \pmatrix{1 & 0 \\ 0 & 1} = \pmatrix{a \\ b}\pmatrix{c & d} = \pmatrix{ac & da \\ bc & bd}. $$ You see that $ac \neq 0$ and that $bd \neq 0$, so $a, b, c, d\neq 0$. So $da\neq 0$ and $bc\neq 0$. (I assume that your constant is zero, otherwise you would just take that constant to be $M$ and $x$ and $y$ both zero vectors.)

share|cite|improve this answer

No, simply because the space of matrices is $mn$-dimensional, and the space of pairs of vectors is $(m+n)$-dimensional, which can be much smaller. The best thing one can do is decompose into a sum of $\min(m,n)$ products, and this decomposition is of course not even close to being unique.

share|cite|improve this answer
Thank you for your answer. What if we have $M_{m,m}$ is a symmetric matrix, which I believe we can write it as $$$$ – user36624 Mar 10 '13 at 1:34

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.