Jx = (1/(Sqrt[2])) {{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}

MatrixForm[Jx]

Eigenvalues[Jx]

losvectores = Eigenvectors[Jx]

v1 = losvectores[[1]]
v2 = losvectores[[2]]
v3 = losvectores[[3]]

u1 = 
      v1/(Sqrt[(v1[[1]]^2 + v1[[2]]^2 + v1[[3]]^2)])
u2 = 
      v2/(Sqrt[(v2[[1]]^2 + v2[[2]]^2 + v2[[3]]^2)])
u3 = 
      v3/(Sqrt[(v3[[1]]^2 + v3[[2]]^2 + v3[[3]]^2)])

MatrixForm[Jx . u1]
MatrixForm[Jx . u2]
MatrixForm[Jx . u3]

u1 . u1
u2 . u2
u3 . u3
u1 . u2
u1 . u3
u2 . u3