Why the standard way to describe rotational inertia of a body is to specify its moment of inertia matrix?
Moment of inertia matrices are somewhat weird: not every symmetric semipositive-definite matrix is a valid moment of inertia matrix (note that there can be no body that has nonzero moment of inertia about exactly one of its principal axes).
Moment of inertia matrix is expected to satisfy I_{around e} = e^T*I*e [0]. At the same time I_{around e} = \sum m_i*r_{perp to e}^2 = \sum m_i*(r_{b1}^2+r_{b2}^2) where b1 and b2 are some orthogonal basis of the surface perpendicular to e.
This creates a natural idea: if we define J := \sum m_i*r_i^T*r_i, then I = C^T*J*C (see [1] for value of C), _and_ every semipositive definite J corresponds to an object that could possibly exist.
So, why don't we use this J instead of I? I think it is less confusing, and seems to be way better e.g. if we're numerically trying to find a moment of inertia that optimizes for something.
[0] So I = \sum m_i*||r_i||^2*Proj_{perp to r_i}^T*Proj_{perp to r_i}
[1] C = \sum_{i != j}e_i^T*e_j (btw. it's not immediately obvious to me that this definition is invariant wrt orthonormal base change, and if you have a succinct description of why it is so, I'd appreciate seeing it)
I was slightly wrong. In actuality, \(I = C_1^T*J*C_1 + C_2^T*J*C_2\) for some \(C_{1,2}\). I should write it up fully, which I'll do soon (tm).
An easy way to see why the original was wrong is that this is the difference between \((a^T+b^T)M(a+b)\) and \(a^TMa+b^TMb\). I could have also been clued in by the impossible properties I claimed C has in [1].