Factoring:
Given variables a, b, c, d, u, v, w, x  in C
Given equations
 a^2 + b^2 = 1
 c^2 + d^2 = 1
 u = ac           the second value, c is complex conjugate of c
 v = ad                             d                         d
 w = bc
 x = bd
then u^2 + v^2 + w*2 + x^2 = 1

given values of a, b, u, v, w, z find values of c and d
if a != 0  then   c = u/a   d = v/a
if b != 0  then   c = w/b   d = x/b  
                  must be at least one because of equation 1

In vector notation, all entries complex numbers:
  |a| outer product |c| = |ac| = |u| again "c" is complex conjugate of c
  |b|               |d|   |ad|   |v|
                          |bc|   |w|
                          |bd|   |x|

In quantum notation:                  c and d conjugates after =
  a|0>+b|1> outer product c|0>+d|1> = ac|00> + ad|01> + bc|10> + dd|11>

Thus when  |p> |q> = |pq>
Given |p> and |pq> then |q> can be determined, factored.

With three vectors                          c, d, e, f conjugates after =
 |a| outter product |c| outer product |e| = |ace| 
 |b|                |d|               |f|   |acf|
                                            |ade|
                                            |adf|
                                            |bce|
                                            |bcf|
                                            |bde|
                                            |bdf|

Given a, b, c, d and 8 element vector, compute e and f

if a != 0   ce=ace/a, cf=acf/a, de=ade/a, df=adf/a 
if b != 0   ce=bce/b. cf=bcf/b, de=bde/b, df=bdf/b

if c !=0    e=ce/c, f=cf/c
if d !=0    e=de/d, f=df/f

etc. for more vectors.

Now consider a unitary 4 by 4 matrix M.
M times conjugate transpose M = I identity matrix.
M is scaled such that the maximum eigen value is less than or equal 1
and the minimum eigen value is greater than or equal -1, 
all entries have magnitude less than or equal 1.

M * |ac ad bc bd| = |u|
                    |v|
                    |w|
                    |x|

Given: a^2+b^2=1, c^2+d^2=1, e^2+f^1=1

Given values a, b, u, v, w, x
find e and f such that
 u = ae
 v = af
 w = be
 x = bf

Use method above.