First consider a rigid body with most of its mass at +1 and 1 on the y plane.
Ixx = sum (y^2)dm = 1+1=2
Iyy = sum(x^2) dm = 0
Ixy = sum(x*y)dm =0
det[M] = Ixx Iyy  Ixy Ixy = 00 =0
(Ixx  )*x = Ixy * y
x/y = Ixy /(Ixx  )
for =0
x/y = 0/2=0
normalising gives a unit vector in the x direction {1,0}T
for =2
x/y = 0/0=?
using the bottom equation
(Iyy  )*y = Ixy * x
y/x = Ixy/(Iyy  )
for =0
y/x = 0/0 = 0
for =2
y/x = 0/(02)=0.5
normalising gives a unit vector in the y direction {0,1}T
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