"Exercise 6.9"

u = (1,0)
d = (0,1)

uu = kronecker(u,u)
ud = kronecker(u,d)
du = kronecker(d,u)
dd = kronecker(d,d)

sigmaz = ((1,0),(0,-1))
sigmax = ((0,1),(1,0))
sigmay = ((0,-i),(i,0))

tauz = ((1,0),(0,-1))
taux = ((0,1),(1,0))
tauy = ((0,-i),(i,0))

I = ((1,0),(0,1))

sigmaz = kronecker(sigmaz,I)
sigmax = kronecker(sigmax,I)
sigmay = kronecker(sigmay,I)

tauz = kronecker(I,tauz)
taux = kronecker(I,taux)
tauy = kronecker(I,tauy)

S = (ud - du) / sqrt(2) -- singlet state

T1 = (ud + du) / sqrt(2)
T2 = (uu + dd) / sqrt(2)
T3 = (uu - dd) / sqrt(2)

"Verify eigenvalues"

check(dot(sigmax,taux,S) == -S)
check(dot(sigmay,tauy,S) == -S)
check(dot(sigmaz,tauz,S) == -S)

check(dot(sigmax,taux,T1) == T1)
check(dot(sigmay,tauy,T1) == T1)
check(dot(sigmaz,tauz,T1) == -T1)

check(dot(sigmax,taux,T2) == T2)
check(dot(sigmay,tauy,T2) == -T2)
check(dot(sigmaz,tauz,T2) == T2)

check(dot(sigmax,taux,T3) == -T3)
check(dot(sigmay,tauy,T3) == T3)
check(dot(sigmaz,tauz,T3) == T3)

H = dot(sigmax,taux) + dot(sigmay,tauy) + dot(sigmaz,tauz)

check(dot(H,S) == -3 S)
check(dot(H,T1) == T1)
check(dot(H,T2) == T2)
check(dot(H,T3) == T3)

"ok"