# Cannot solve system of simultaneous equations

I am trying to solve a particular problem, but my test solution for setting up the code isn't working. I have tried two ways - one, which just outputs the same equations, and the other, I'm unsure on how to solve this large list of equations which are unrealistic to type out.

First attempt

```
var('a b c d e f g h i')
X=matrix(3,3,[[0,1,0],[0,0,1],[1,0,0]])
P=matrix(3,3,[[a,b,c],[d,e,f],[g,h,i]])
Pdagger=P.transpose()
Xdagger=X.transpose()
L=P*X*Pdagger
```

This outputs $$\begin{pmatrix} ab+ac+bc & cd+ae+bf & cg+ah+bi \\ bd+ce+af & de+df+ef & fg+dh+ei \\ bg+ch+ai & eg+fh+di & gh+gi+hi \end{pmatrix}$$ I then make $L=X$, then type these equations element wise into the simultaneous equation solver;

```
equations=solve([a*b+a*c+b*c==0,c*d+a*e+b*f==1,c*g+a*h+b*i==0,b*d+c*e+a*f==0,d*e+d*f+e*f==0,f*g+d*h+e*i==1,b*g+c*h+a*i==1,e*g+f*h+d*i==0,g*h+g*i+h*i==0],a,b,c,d,e,f,g,h,i)
```

The output:

```
a*e + b*f == 1, d*e + d*f + e*f == 0, b*g + c*h + a*i == 1, c*g + a*h + b*i == 0, e*g + f*h + d*i == 0, f*g + d*h + e*i == 1, g*h + g*i + h*i == 0]
```

Second attempt;

```
Q=X*Pdagger
B=Q.solve_left(X)
eqn=[]
for i in range(0,3):
for j in range(0,3):
eqn.append(B[i][j])
```

This gives me a list of the equations, I then try the solver again;

```
solve([eqn[0]==0,...eqn[8]==0],a,b,c,d,e,f,g,h,i)
```

This just gives me an error:

```
(a, b, c, d, e, f, g, h, i)
Error in lines 13-13
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute
exec compile(block+'\n', '', 'single') in namespace, locals
File "", line 1, in <module>
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/relation.py", line 976, in solve
raise TypeError("%s is not a valid variable." % repr(i))
TypeError: 2 is not a valid variable.
```

The hope is that I will be able to extend this to larger matrices, so autonomy is welcomed greatly. But I will happily take any advice given I'm at a completely lose end. It is worth noting that the solution to this problem; $$PXP^{\dagger}=X$ is the identity matrix. My understanding is that the first method cannot find a closed form? I have even tried specifying that a,b,c,d,e,f,g,h,i are all 0 or 1 and this has not helped. I believe my second method is more on the right track, however I cannot afford to manually type out the 9 non-linear equations that pop out from the matrix solver.