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 2018-04-27 09:03:28 -0500 received badge ● Student (source) 2018-04-26 04:38:43 -0500 asked a question Sage wont solve simultaneous equation I tried asking this question earlier, but my profile still says I have asked 0 questions. I have tried solving this problem in different ways, but this is the closest I have gotten; 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() Q=X*Pdagger B=Q.solve_left(X) eqns=[] xlist=[] for i in range(0,3): for j in range(0,3): eqns.append(B[i][j]) xlist.append(X[i][j]) final=[] for k in range(0,9): final.append(eqns[k]==xlist[k]) show(final) sol=solve(eqns,[a,b,c,d,e,f,g,h,i]) show(sol)  I know there is a solution to this problem, and it is the identity, $PXP^{\dagger}=X$. But the output: Error in lines 19-19 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 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.  There is also probably a much more efficient way of specifying the problem, but I don't understand why this wont work. I would like to extend this problem to larger matrices as well, so autonomy is welcomed. However, I would appreciate any advice just on getting this error fixed. 2018-04-26 04:38:43 -0500 asked a question 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 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; $$PXP^{\dagger}=X is the identity matrix. My understanding is that the first method cannot find a closed form? I have even tried specifying that a,b,c,d,e,f,g,h,i are all 0 or 1 and this has not helped. I believe my second method is more on the right track, however I cannot afford to manually type out the 9 non-linear equations that pop out from the matrix solver. 2018-04-26 04:38:43 -0500 asked a question How to solve a large list of simultaneous equations I am trying to solve a particular problem, but my test solution for setting up the code isn't working. I have tried two ways - one, which just outputs the same equations, and the other, I'm unsure on how to solve this large list of equations which are unrealistic to type out. First attempt var('a b c d e f g h i') X=matrix(3,3,[[0,1,0],[0,0,1],[1,0,0]]) P=matrix(3,3,[[a,b,c],[d,e,f],[g,h,i]]) Pdagger=P.transpose() Xdagger=X.transpose() L=P*X*Pdagger  This outputs$$\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}$$I then make L=X, then type these equations element wise into the simultaneous equation solver; equations=solve([a*b+a*c+b*c==0,c*d+a*e+b*f==1,c*g+a*h+b*i==0,b*d+c*e+a*f==0,d*e+d*f+e*f==0,f*g+d*h+e*i==1,b*g+c*h+a*i==1,e*g+f*h+d*i==0,g*h+g*i+h*i==0],a,b,c,d,e,f,g,h,i)  The output:  a*e + b*f == 1, d*e + d*f + e*f == 0, b*g + c*h + a*i == 1, c*g + a*h + b*i == 0, e*g + f*h + d*i == 0, f*g + d*h + e*i == 1, g*h + g*i + h*i == 0]  Second attempt; Q=X*Pdagger B=Q.solve_left(X) eqn=[] for i in range(0,3): for j in range(0,3): eqn.append(B[i][j])  This gives me a list of the equations, I then try the solver again; solve([eqn[0]==0,...eqn[8]==0],a,b,c,d,e,f,g,h,i)  This just gives me an error: (a, b, c, d, e, f, g, h, i) Error in lines 13-13 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/relation.py", line 976, in solve raise TypeError("%s is not a valid variable." % repr(i)) TypeError: 2 is not a valid variable.  The hope is that I will be able to extend this to larger matrices, so autonomy is welcomed greatly. But I will happily take any advice given I'm at a completely lose end. It is worth noting that the solution to this problem;$$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.