Why, given an object with rotational symmetry, is the axis of symmetry a principal axis?

Question

When consulting textbooks and notes online about principle axes of inertia, I couldn't find a source which directly addressed the reasoning/proof behind following statement:

"Given an object with a rotational symmetry, the axis of symmetry is a principal axis."

I can only find (many) examples where this property has been used to find one of the principal axes. It seems to be so widely accepted, as almost a trivial thing, but

1) what is the intuition behind the statement,

2) as well as a more rigorous treatment.

Answer 1

Any symmetry of the object is also a symmetry of the eigenspaces. Hence the only possibilities are

• One 1-dimensional eigenspace along the axis, one 2-dimensional eigenspace orthogonal to it
• One 3-dimensional eigenspace (i.e., there are not really any preferred prnciple axes, so we might take the rotational axis as one anyway)