How to do rank-1 factorization in MATLAB? -

i have matrix m of dimensions 6x6 , has rank 1. how can factorize 2 matrices of dimensions 6x1 (say a) , 1x6 (say b) m=a*b.

take largest eigen vector , multiply largest eigenvalue :

 a=[1 2 3 4]'*[1 2 3 4]  =       1     2     3     4      2     4     6     8      3     6     9    12      4     8    12    16   [v,e] = eigs(a,1);  sqrt(e)*v  ans =     -1.0000    -2.0000    -3.0000    -4.0000 

of course, result sign change.

edit: if assume 2 vectors can different:

a=[1 2 3 4]'*[5 6 7 8] [uu,ss,vv]=svd(a); u=uu(:,1)*ss(1,1) v=vv(:,1) assert(norm(u*v'-a)<1e-10) 

now solution less unique. determining 2*n values based on n. 1 solution among many.

for example, @ other simpler solution (which assume matrix rank 1) :

aa=a(:,:)./repmat(a(1,:),[size(a,1),1]); bb=a(:,:)./repmat(a(:,1),[1,size(a,2)]); u=aa(:,1); v=bb(1,:)'*a(1); assert(norm(u*v'-a)<1e-10) 

it produces totally different result, still factorizes matrix. if want non-negative factorizations reduce space of possible results, i'd suggest ask new question!