So it's easy to take the tensor product of two vectors, [ a b ] ⊗ [ c d ] = [ a c a d b c b d ] but it seems much more difficult to go backwards, to "factor" the vector. Is there a method to find two vectors whose tensor product is a given vector?In my specific problem, the magnitude of the resultant vector is 1, if that's relevant.

Question

So it&#039;s easy to take the tensor product of two vectors,      [                            a                                      b                      ]    ⊗      [                            c                                      d                      ]    =      [                            a          c                                      a          d                                      b          c                                      b          d                      ]  but it seems much more difficult to go backwards, to &quot;factor&quot; the vector. Is there a method to find two vectors whose tensor product is a given vector?In my specific problem, the magnitude of the resultant vector is 1, if that&#039;s relevant.

Jonathan Bailey · Accepted Answer

To find a solution, assume your tensor is   (  x  ,  y  ,  z  ,  w      )    ⊤    .. We must have   x  w  =  a  b  c  d  =  y  z for there to be any solution.Assume   x  ≠  0.Then      b    a    =      z    x          d    c    =      y    x    .So you get that      (                            1                                      z                      /                    x                      )    ⊗      (                            1                                      y                      /                    x                      )    =      (                            1                                      y                      /                    x                                      z                      /                    x                                      y          z                      /                                x            2                                )    =      (                            1                                      y                      /                    x                                      z                      /                    x                                      w                      /                    x                      )    =      1    x        (                            x                                      y                                      z                                      w                      )    ,With one step because yz=xw.So we can choose   a  =  1  ,  b  =  z      /    x  ,  c  =  x  ,  d  =  y  .More generally, if uv=x, we can choose:  a  =  u  ,  b  =  u  z      /    x  =  z      /    v  ,  c  =  v  ,  d  =  v  y      /    x  =  y      /    u  .If x=0, you can do the same with any other non-zero coordinate.If they are all zero, you can set a=b=0 and c,d to anything, or c=d=0 and a,b anything.

So it's easy to take the tensor product of two vectors, [[a],[b]] ox [[c],[d]]=[[ac],[ad],[bc],[bd]] but it seems much more difficult to go backwards, to "factor" the vector. Is there a method to find two vectors whose tensor product is a given vector?

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