with(LinearAlgebra):

# primitive root of unity
w := (sqrt(5)-1)/4 + I*sqrt((5+sqrt(5))/8): 

u1 := -5/2 - I/2*sqrt(25-10*sqrt(5)):
u2 := (-5-5*sqrt(5) - I*sqrt(10*(5-sqrt(5))))/4:
u3 := (5-5*sqrt(5) + I*sqrt(10*(5+sqrt(5))))/4:

# original matrix
W := <<1,1,1,1,1>|<1,w,w^2,w^3,w^4>|<1,w^2,w^4,w,w^3>|<1,w^3,w,w^4,w^2>|<1,w^4,w^3,w^2,w>>:

# 3 nonzero entries
S := <<u1,0,0,0,0>|<0,u2,0,0,0>|<0,0,u3,0,0>|<0,0,0,0,0>|<0,0,0,0,0>>:

# compute rank
printf("The rank of W+S is: %d\n", Rank(W+S));