# Does moment of inertia only work for special cases?

I was looking into the moment of inertia expression for angular momentum. The angular momentum of a group of particles can be expressed as a linear transformation of the angular velocity vector. This produces a simple form for the whole body when the angular velocity is constant. Moreover, assuming we would like a simpler diagonal form for moment of inertia, the matrix must be diagonalisable so also must have real eigenvalues. The rotation vector must also be parallel to an eigenvector if we further want a constant moment of inertia

My question is, are these conditions necessary for moment of inertia to make sense? Moreover what shortcuts do we know a priori of calculating the linear transformation, to tell us that these conditions do in fact hold?

Constant angular velocity, for I to exist Diagonalisable Matrix with angular velocity parallel to one eigenvector

If so, what are some shortcuts we can take to observe when these conditions hold?

Sorry if my question was unclear