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When I try to solve (for complex c[i,j])

eq= (15 Sqrt[6] + 568 c[1, 1] Conjugate[c[2, 1]] == 
  1704 Abs[c[1, 1]]^2 && (45 Sqrt[3/2])/568 + Abs[c[2, 1]]^2 + 
   Abs[c[2, 2]]^2 == 3 c[2, 1] Conjugate[c[1, 1]]);

by running

sol=Solve[eq, {c[1, 1], c[2, 2]}]

I get the error

Inconsistent or redundant transcendental equation. After reduction, the bad equation is -15 Sqrt[6]+1704 c[1,1]^2-568 c[1,1] Conjugate[c[2,1]] == 0.

I also find several solutions. What is the meaning of the warning? I find it confusing. For one thing inconsistent and redundant seem to be quite the opposite (one implying no solution exists the other implying that an equation is redundant so that there should be one constraint less). Acting with Solve on the bad equation gives a solution.

Plugging in a solution eq \. sol and substituting a value for c[2,1] (since the remaining equation is still non-trivial) I find False. So I guess that I didn't really find a solution.

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  • $\begingroup$ Could try sol = Solve[eq, {c[1, 1], c[2, 2]}, Method -> Reduce]. Also note that the warning has an "or" in it. The internal equations encountered either are inconsistent OR they are redundant. $\endgroup$ Jun 20, 2022 at 18:51

1 Answer 1

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One way is:

eq= (15 Sqrt[6] + 568 c[1, 1] Conjugate[c[2, 1]] == 
1704 Abs[c[1, 1]]^2 && (45 Sqrt[3/2])/568 + Abs[c[2, 1]]^2 + 
Abs[c[2, 2]]^2 == 3 c[2, 1] Conjugate[c[1, 1]]);
sol = Reduce[eq, {c[1, 1], c[2, 2]}]

(*Long answer*)
 

Or using: $$z^*=\frac{| z| ^2}{z}$$

 eq = (15 Sqrt[6] + 568 c[1, 1]*Abs[c[2, 1]]^2/c[2, 1] == 
 1704 Abs[c[1, 1]]^2 && (45 Sqrt[3/2])/568 + Abs[c[2, 1]]^2 + 
  Abs[c[2, 2]]^2 == 3 c[2, 1] Abs[c[1, 1]]^2/c[1, 1]) // Simplify

 (*15 Sqrt[6] + (568 Abs[c[2, 1]]^2 c[1, 1])/c[2, 1] == 
  1704 Abs[c[1, 1]]^2 && (45 Sqrt[3/2])/568 + Abs[c[2, 1]]^2 + 
 Abs[c[2, 2]]^2 == (3 Abs[c[1, 1]]^2 c[2, 1])/c[1, 1]*)

 Solve[eq, {c[1, 1], c[2, 2]}] // Simplify

 (*{{c[1, 1] -> -(1/6) Sqrt[(45 Sqrt[3/2])/71 + Abs[c[2, 1]]^4/
  c[2, 1]^2] + Abs[c[2, 1]]^2/(6 c[2, 1]), 
  c[2, 2] -> -(1/4) Sqrt[-((45 Sqrt[6])/71) - 8 Abs[c[2, 1]]^2 - 
 4 Sqrt[(90 Sqrt[6])/71 + (4 Abs[c[2, 1]]^4)/c[2, 1]^2]
   c[2, 1]]}, {c[1, 1] -> -(1/6) Sqrt[(45 Sqrt[3/2])/71 + Abs[c[2, 1]]^4/c[2, 1]^2] + Abs[c[2, 1]]^2/(6 c[2, 1]), 
  c[2, 2] -> 1/4 Sqrt[-((45 Sqrt[6])/71) - 8 Abs[c[2, 1]]^2 - 
  4 Sqrt[(90 Sqrt[6])/71 + (4 Abs[c[2, 1]]^4)/c[2, 1]^2]
   c[2, 1]]}, {c[1, 1] -> 
   1/6 Sqrt[(45 Sqrt[3/2])/71 + Abs[c[2, 1]]^4/c[2, 1]^2] + 
   Abs[c[2, 1]]^2/(6 c[2, 1]), 
   c[2, 2] -> -(1/4) Sqrt[-((45 Sqrt[6])/71) - 8 Abs[c[2, 1]]^2 + 
   4 Sqrt[(90 Sqrt[6])/71 + (4 Abs[c[2, 1]]^4)/c[2, 1]^2]
   c[2, 1]]}, {c[1, 1] -> 
1/6 Sqrt[(45 Sqrt[3/2])/71 + Abs[c[2, 1]]^4/c[2, 1]^2] + 
Abs[c[2, 1]]^2/(6 c[2, 1]), 
c[2, 2] -> 
1/4 Sqrt[-((45 Sqrt[6])/71) - 8 Abs[c[2, 1]]^2 + 
 4 Sqrt[(90 Sqrt[6])/71 + (4 Abs[c[2, 1]]^4)/c[2, 1]^2] c[2, 1]]}}*)
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