Moment of Inertia Matrix

1. Moment of Inertia Matrix Angular Momentum: Matrix Equation:

2. Moment of Inertia Matrix Solid objects: Symmetric Matrix: • Matrix has real eigenvalues • Matrix has orthogonal eigenvectors • Can be rotated into diagonal form

3. Rotation to Diagonal Form Eigenvectors for a diagonal matrix: Eigenvalues Eigevalue same for all axes Eigenvector depends on axes

4. Eigenvalues Characteristic equation determines eigenvalues Each eigenvalue Eigenvector

5. Eigenvectors Are Perpendicular

6. Rotation to Diagonal Form The matrix that rotates to diagonal form is:

7. Moment of Inertia for Pyramid Eigenvalues: Eigenvectors:

8. Diagonalize Matrix? Not diagonal! But we have redundant eigenvalues! That means any linear combination of eigenvectors work equally well.

9. Parallel Axis Theorem 0