Moments of Inertia: Finding the Eigenvalues of a Matrix

• We look at a rigid body with moments of inertia 1, 3, 5 (in suitable units). In a special coordinate system it is described by the diagonal matrix

      J = diag([1 3 5])
     
     J =
     
          1     0     0
          0     3     0
          0     0     5

• We rotate the system, first by 30 degrees around the x axes, then 45 degrees around the z axes, using the rotation matrices

      ang1 = 30/180*pi
     
     ang1 =
     
         0.5236

      Q1 = [1 0 0; 0 cos(ang1) sin(ang1) ; 0 -sin(ang1) cos(ang1)]
     
     Q1 =
     
         1.0000         0         0
              0    0.8660    0.5000
              0   -0.5000    0.8660
     
      ang2 = 45/180*pi
     
     ang2 =
     
         0.7854
     
      Q2 = [cos(ang2) sin(ang2) 0; -sin(ang2) cos(ang2) 0; 0 0 1]
     
     Q2 =
     
         0.7071    0.7071         0
        -0.7071    0.7071         0
              0         0    1.0000

• This leads to the following matrix for the moment of inertia in the rotated system:

      Jrot = Q2*(Q1*J*Q1')*Q2'
     
     Jrot =
     
         2.2500    1.2500    0.6124
         1.2500    2.2500    0.6124
         0.6124    0.6124    4.5000

• The principal moments of inertia are given by the eigenvalues of Jrot:

      eig(Jrot)
     
     ans =
     
          1
          3
          5

• which of course was expected.