I have some problems to simplify 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 must be unitary, so there are conditions on the c. I declare my conditions as

`cond = {{(Abs[c[c[c[c[1, 1]]^ 2 + Abs[c[c[c[c[1, 2]]^ 2 + Abs[c[c[c[c[1, 3]]^ 2) -> 1, (c[1, 2] Conjugate[c[c[c[c[1, 1]]+ c[2, 2] Conjugate[c[c[c[c[1, 2]]+ c[2, 3] Conjugate[c[c[c[c[1, 3]]) -> 0, (c[1, 3] Conjugate[c[c[c[c[1, 1]]+ c[2, 3] Conjugate[c[c[c[c[1, 2]]+ c[3, 3] Conjugate[c[c[c[c[1, 3]]) -> 0, (c[1, 1] Conjugate[c[c[c[c[1, 2]]+ c[1, 2] Conjugate[c[c[c[c[2, 2]]+ c[1, 3] Conjugate[c[c[c[c[2, 3]]) -> 0, (Abs[c[c[c[c[1, 2]]^ 2 + Abs[c[c[c[c[2, 2]]^ 2 + Abs[c[c[c[c[2, 3]]^ 2) -> 1, (c[1, 3] Conjugate[c[c[c[c[1, 2]]+ c[2, 3] Conjugate[c[c[c[c[2, 2]]+ c[3, 3] Conjugate[c[c[c[c[2, 3]]) -> 0, (c[1, 1] Conjugate[c[c[c[c[1, 3]]+ c[1, 2] Conjugate[c[c[c[c[2, 3]]+ c[1, 3] Conjugate[c[c[c[c[3, 3]]) -> 0, (c[1, 2] Conjugate[c[c[c[c[1, 3]]+ c[2, 2] Conjugate[c[c[c[c[2, 3]]+ c[2, 3] Conjugate[c[c[c[c[3, 3]]) -> 0, (Abs[c[c[c[c[1, 3]]^ 2 + Abs[c[c[c[c[2, 3]]^ 2 + Abs[c[c[c[c[3, 3]]^ 2-> 1)}}`

Then I want to diagonalize the next matrix `S = 1/3 {{-1, 2, 2}, {2, -1, 2}, {2, 2, -1}}`

and I want the diagonal matrix to have the shape `diagS = DiagonalMatrix[{1,-1,-1}]`

so I use

`FullSimplify[ConjugateTranspose[W].S.W - DiagonalMatrix[{1, -1, -1}],`

/.Flatten[cond]]

I receive

`{{1/3 (-3 - Abs[c[c[c[c[1, 1]]^ 2 - Abs[c[c[c[c[1, 2]]^ 2 - Abs[c[c[c[c[1, 3]]^ 2 + 2 (c[1, 2] + c[1, 3]) Conjugate[c[c[c[c[1, 1]]+ 2 (c[1, 1] + c[1, 3]) Conjugate[c[c[c[c[1, 2]]+ 2 (c[1, 1] + c[1, 2]) Conjugate[c[c[c[c[1, 3]]), 1/3 ((-c[1, 2] + 2 (c[2, 2] + c[2, 3])) Conjugate[c[c[c[c[1, 1]]+ (2 c[1, 2] - c[2, 2] + 2 c[2, 3]) Conjugate[c[c[c[c[1, 2]]+ (2 (c[1, 2] + c[2, 2]) - c[2, 3]) Conjugate[c[c[c[c[1, 3]]), 1/3 ((-c[1, 3] + 2 (c[2, 3] + c[3, 3])) Conjugate[c[c[c[c[1, 1]]+ (2 c[1, 3] - c[2, 3] + 2 c[3, 3]) Conjugate[c[c[c[c[1, 2]]+ (2 (c[1, 3] + c[2, 3]) - c[3, 3]) Conjugate[c[c[c[c[1, 3]])}, {1/3 ((-c[1, 1] + 2 (c[1, 2] + c[1, 3])) Conjugate[c[c[c[c[1, 2]]+ (2 c[1, 1] - c[1, 2] + 2 c[1, 3]) Conjugate[c[c[c[c[2, 2]]+ (2 (c[1, 1] + c[1, 2]) - c[1, 3]) Conjugate[c[c[c[c[2, 3]]), 1/3 (3 - Abs[c[c[c[c[1, 2]]^ 2 - Abs[c[c[c[c[2, 2]]^ 2 - Abs[c[c[c[c[2, 3]]^ 2 + 2 (c[2, 2] + c[2, 3]) Conjugate[c[c[c[c[1, 2]]+ 2 (c[1, 2] + c[2, 3]) Conjugate[c[c[c[c[2, 2]]+ 2 (c[1, 2] + c[2, 2]) Conjugate[c[c[c[c[2, 3]]), 1/3 ((-c[1, 3] + 2 (c[2, 3] + c[3, 3])) Conjugate[c[c[c[c[1, 2]]+ (2 c[1, 3] - c[2, 3] + 2 c[3, 3]) Conjugate[c[c[c[c[2, 2]]+ (2 (c[1, 3] + c[2, 3]) - c[3, 3]) Conjugate[c[c[c[c[2, 3]])}, {1/3 ((-c[1, 1] + 2 (c[1, 2] + c[1, 3])) Conjugate[c[c[c[c[1, 3]]+ (2 c[1, 1] - c[1, 2] + 2 c[1, 3]) Conjugate[c[c[c[c[2, 3]]+ (2 (c[1, 1] + c[1, 2]) - c[1, 3]) Conjugate[c[c[c[c[3, 3]]), 1/3 ((-c[1, 2] + 2 (c[2, 2] + c[2, 3])) Conjugate[c[c[c[c[1, 3]]+ (2 c[1, 2] - c[2, 2] + 2 c[2, 3]) Conjugate[c[c[c[c[2, 3]]+ (2 (c[1, 2] + c[2, 2]) - c[2, 3]) Conjugate[c[c[c[c[3, 3]]), 1/3 (3 - Abs[c[c[c[c[1, 3]]^ 2 - Abs[c[c[c[c[2, 3]]^ 2 - Abs[c[c[c[c[3, 3]]^ 2 + 2 (c[2, 3] + c[3, 3]) Conjugate[c[c[c[c[1, 3]]+ 2 (c[1, 3] + c[3, 3]) Conjugate[c[c[c[c[2, 3]]+ 2 (c[1, 3] + c[2, 3]) Conjugate[c[c[c[c[3, 3]])}}`

Clearly, `FullSimplify`

do not use my conditions since `-Abdos[c[c[c[c[1, 1]]^ 2 - Abs[c[c[c[c[1, 2]]^ 2 - Abs[c[c[c[c[1, 3]]^ 2 = -1`

Maybe I do not use it `FullSimplify`

correctly, then I would be grateful for help. Thank you in advance.