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I have some problems simplifying things with FullSimplify. Suppose a matrix W = {{c[1, 1],c[1, 2],c[1, 3]}, {c[1, 2],c[2, 2],c[2, 3]},{c[1, 3],c[2, 3],c[3, 3]}} which has to be unitary, so there are conditions on the c's. I declare my conditions as
cond ={{(Abs[c[1, 1]]^2+Abs[c[1, 2]]^2+Abs[c[1, 3]]^2)->1,(c[1, 2] Conjugate[c[1, 1]]+c[2, 2] Conjugate[c[1, 2]]+c[2, 3] Conjugate[c[1, 3]])->0,(c[1, 3] Conjugate[c[1, 1]]+c[2, 3] Conjugate[c[1, 2]]+c[3, 3] Conjugate[c[1, 3]])->0,(c[1, 1] Conjugate[c[1, 2]]+c[1, 2] Conjugate[c[2, 2]]+c[1, 3] Conjugate[c[2, 3]])->0,(Abs[c[1, 2]]^2+Abs[c[2, 2]]^2+Abs[c[2, 3]]^2)->1,(c[1, 3] Conjugate[c[1, 2]]+c[2, 3] Conjugate[c[2, 2]]+c[3, 3] Conjugate[c[2, 3]])->0,(c[1, 1] Conjugate[c[1, 3]]+c[1, 2] Conjugate[c[2, 3]]+c[1, 3] Conjugate[c[3, 3]])->0,(c[1, 2] Conjugate[c[1, 3]]+c[2, 2] Conjugate[c[2, 3]]+c[2, 3] Conjugate[c[3, 3]])->0,(Abs[c[1, 3]]^2 + Abs[c[2, 3]]^2+Abs[c[3, 3]]^2->1)}} Then, I want to diagonalize the following matrix S = 1/3 {{-1, 2, 2}, {2, -1, 2}, {2, 2, -1}} and I want that the diagonal matrix has the form diagS=DiagonalMatrix[{1,-1,-1}] so I use
FullSimplify[ConjugateTranspose[W].S.W - DiagonalMatrix[{1, -1, -1}], /.Flatten[cond]]
I get
{{1/3 (-3 - Abs[c[1, 1]]^2 - Abs[c[1, 2]]^2 - Abs[c[1, 3]]^2 + 2 (c[1, 2] + c[1, 3]) Conjugate[c[1, 1]] + 2 (c[1, 1] + c[1, 3]) Conjugate[c[1, 2]] + 2 (c[1, 1] + c[1, 2]) Conjugate[c[1, 3]]),1/3 ((-c[1, 2] + 2 (c[2, 2] + c[2, 3])) Conjugate[c[1, 1]] + (2 c[1, 2] - c[2, 2] + 2 c[2, 3]) Conjugate[c[1, 2]] + (2 (c[1, 2] + c[2, 2]) - c[2, 3]) Conjugate[c[1, 3]]),1/3 ((-c[1, 3] + 2 (c[2, 3] + c[3, 3])) Conjugate[c[1, 1]] + (2 c[1, 3] - c[2, 3] + 2 c[3, 3]) Conjugate[c[1, 2]] + (2 (c[1, 3] + c[2, 3]) - c[3, 3]) Conjugate[c[1, 3]])}, {1/3 ((-c[1, 1] + 2 (c[1, 2] + c[1, 3])) Conjugate[c[1, 2]] + (2 c[1, 1] - c[1, 2] + 2 c[1, 3]) Conjugate[c[2, 2]] + (2 (c[1, 1] + c[1, 2]) - c[1, 3]) Conjugate[c[2, 3]]), 1/3 (3 - Abs[c[1, 2]]^2 - Abs[c[2, 2]]^2 - Abs[c[2, 3]]^2 + 2 (c[2, 2] + c[2, 3]) Conjugate[c[1, 2]] + 2 (c[1, 2] + c[2, 3]) Conjugate[c[2, 2]] + 2 (c[1, 2] + c[2, 2]) Conjugate[c[2, 3]]), 1/3 ((-c[1, 3] + 2 (c[2, 3] + c[3, 3])) Conjugate[c[1, 2]] + (2 c[1, 3] - c[2, 3] + 2 c[3, 3]) Conjugate[c[2, 2]] + (2 (c[1, 3] + c[2, 3]) - c[3, 3]) Conjugate[c[2, 3]])}, {1/3 ((-c[1, 1] + 2 (c[1, 2] + c[1, 3])) Conjugate[c[1, 3]] + (2 c[1, 1] - c[1, 2] + 2 c[1, 3]) Conjugate[c[2, 3]] + (2 (c[1, 1] + c[1, 2]) - c[1, 3]) Conjugate[c[3, 3]]),1/3 ((-c[1, 2] + 2 (c[2, 2] + c[2, 3])) Conjugate[c[1, 3]] + (2 c[1, 2] - c[2, 2] + 2 c[2, 3]) Conjugate[c[2, 3]] + (2 (c[1, 2] + c[2, 2]) - c[2, 3]) Conjugate[c[3, 3]]),1/3 (3 - Abs[c[1, 3]]^2 - Abs[c[2, 3]]^2 - Abs[c[3, 3]]^2 + 2 (c[2, 3] + c[3, 3]) Conjugate[c[1, 3]] + 2 (c[1, 3] + c[3, 3]) Conjugate[c[2, 3]] + 2 (c[1, 3] + c[2, 3]) Conjugate[c[3, 3]])}} Clearly, FullSimplify is not using my conditions since -Abs[c[1, 1]]^2 - Abs[c[1, 2]]^2 - Abs[c[1, 3]]^2 = -1
Maybe I am not using FullSimplify correctly, so I would appreciate some help. Thanks in advance.

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  • $\begingroup$ Why not try in a simpler setting first? Work with 2D matrices where things are simpler, get that functioning, and then expand to your real problem. $\endgroup$ – bill s Jun 26 at 14:08
  • $\begingroup$ Maybe read about ComplexExpand. $\endgroup$ – Marius Ladegård Meyer Jun 28 at 8:36

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