How to find the original vector given the outer product of the vector?

Suppose we have a vector:

psi = rand(n,1) + 1i * rand(n,1);

and we compute its outer product:

outer_product = psi * psi'

Now, given this outer_product, is there a way to retrieve the original vector?

Answer 1

[Matt J]

It's only possible up to a multiplicative constant of the form .

n=5; psi0 = rand(n,1) + 1i * rand(n,1);

outer_product = psi0 * psi0';

[psi,s]=svds(outer_product,1);

psi=psi*sqrt(s); %recovered psi

abs(psi./psi0)

ans = 5×1

1.0000 1.0000 1.0000 1.0000 1.0000

Answer 2

[Steven Lord]

n = 3;
psi1 = rand(n,1) + 1i * rand(n,1)

psi1 = 
   0.2707 + 0.4633i
   0.8235 + 0.2196i
   0.4234 + 0.1861i

outer_product1 = psi1 * psi1'

outer_product1 = 
   0.2879 + 0.0000i   0.3247 + 0.3221i   0.2008 + 0.1458i
   0.3247 - 0.3221i   0.7264 + 0.0000i   0.3895 - 0.0603i
   0.2008 - 0.1458i   0.3895 + 0.0603i   0.2139 + 0.0000i

psi2 = 1i*psi1

psi2 = 
  -0.4633 + 0.2707i
  -0.2196 + 0.8235i
  -0.1861 + 0.4234i

outer_product2 = psi2 * psi2'

outer_product2 = 
   0.2879 + 0.0000i   0.3247 + 0.3221i   0.2008 + 0.1458i
   0.3247 - 0.3221i   0.7264 + 0.0000i   0.3895 - 0.0603i
   0.2008 - 0.1458i   0.3895 + 0.0603i   0.2139 + 0.0000i

norm(outer_product1-outer_product2)

ans = 1.3432e-16

outer_product1 and outer_product2 are effectively identical. So if I gave you that outer_product, did I generate it using psi1 or psi2?