Matrix distribution into two vectors that build similiar matrix as a vector product

Hello,

i want to split up a matrix into two vectors. If i build the vector product of those vectors a similiar matrix compared to the original one should emerge. This should be an optimization problem, but i don't know where to start.

Furthermore the solution should not be an iterative one.

Thank you guys a lot!

Answer 1

[Torsten]

But you already got the answer for the two vectors a and b and the resulting approximating matrix C:

[u s v] = svd(M);
n = 1;                  % initially
a = n*u(:,1);
b = (s(1,1)/n)*v(:,1);
C = a*b'

[Henning Schmitz]

No this does not work unfortunately for a random matrix.

for example:

M2D = rand(4,4)
  n = 5;  
[u s v] = svd(M2D);


ax_test = n*u(:,1);
ay_test = (s(1,1)/n)*v(:,1);
M2D_test = ax_test*ay_test';

It just works (very good) if the matrix (M2D) is made as product of two vectors.

[Torsten]

It's the best reconstruction for M2D that you can get from two vectors. You can't expect a perfect equality (M2D = a*b') since rank(a*b') = 1 whereas rank(M2D) = 4 usually.