Is it possible to decompose a matrix as the product of two vectors?

Question

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 $\vec{x}$?

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.) — , Mar 10 '13 at 02:02
possible duplicate of Factorize a Symmetric matrix as an 'Approximation' with an outer product. — Tim Seguine, May 25 '14 at 22:05
@TimSeguine That question is specific to symmetric matrices, and the answer there uses that assumption. Here we have a rectangular matrix. — , May 25 '14 at 22:26
@wordsthatendinGRY I get the impression from the comment from the OP to the top answer here, that they are interested precisely in symmetric matrices. If that is not the case, then you are quite correct. — Tim Seguine, May 25 '14 at 23:10

score 7 · Answer 1 · answered Mar 10 '13 at 01:25

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.)

score 7 · Answer 2 · answered Mar 10 '13 at 02:52

7

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.

answered Mar 10 '13 at 02:52

Learner

2,696

score 6 · Answer 3 · answered Mar 10 '13 at 01:22

6

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.

answered Mar 10 '13 at 01:22

Alexander Shamov

4,457
17
28

Thank you for your answer. What if we have $M_{m,m}$ is a symmetric matrix, which I believe we can write it as $$$$ – pitfall Mar 10 '13 at 01:34

score 1 · Answer 4 · answered Sep 10 '22 at 09:27

Let us think of this as trying to solve for two vectors $a_{m \times 1},b_{n \times 1}$ given a matrix $A_{m\times n}$ such that $ab^T = A$

Note that the number of variables you have are number of elements in $a$ + number of elements in $b$

$ab^{T}$ gives you a matrix $B_{m \times n}$ with $m\times n$ elements.

$A = B$ if every element of $A$ = every element of $B$.

This means that you need to satisfy $m \times n$ equation with $m+n$ variables. This system doesn't necessarily have a solution unless there are constraints on the elements of $A$ itself.

In general $m \times n \neq (m+n)$

Even when $m+n = mn$ it is not guaranteed that this problem can be solved. consider $A$ to be $2 \times 2 $. You run into issues when one of the entries is zero as shown in one of the above answers.

This was just to give an intuitive picture as to why you can't always express a matrix as a product of two 1-dimensional matrices.

Is it possible to decompose a matrix as the product of two vectors?

4 Answers4

Linked