Terminologies for moment of inertia Perhaps someone can suggest the right terms for the following mathematical objects related to moment of inertia?


*

*A inertia tensor $I$. $$I \equiv \begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} \\ I_{2,1} & I_{2,2} & I_{2,3} \\ I_{3,1} & I_{3,2} & I_{3,3} \\\end{bmatrix}$$

*A product of inertia is an off-diagonal entry in the tensor: $I_{1,2} = I_{2,1}$, $I_{1,3} = I_{3,1}$, or $I_{2,3} = I_{3,2}$.

*A principal moment of inertia is a diagonal entry in the tensor: $I_{1,1}$, $I_{2,2}$, or $I_{3,3}$. This is the semantic of moment of inertia discussed in elementary treatment of Physics.

*What is the term for $I_2$ and $I_3$ in the last line below? $$\begin{align*}
          I_{1,1}
          &= \sum_{j} m_j\;\left(r^2_{j,2} + r^2_{j,3}\right) \\
          &= \sum_{j} m_j\,r^2_{j,2} + \sum_{j} m_j\,r^2_{j,3} \\
          I_{1,1} &= I_2 + I_3
        \end{align*}$$

 A: 
A inertia tensor $I$. $$I \equiv \begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} \\ I_{2,1} & I_{2,2} & I_{2,3} \\ I_{3,1} & I_{3,2} & I_{3,3} \\\end{bmatrix}$$
    A product of inertia is an off-diagonal entry in the tensor: $I_{1,2} = I_{2,1}$, $I_{1,3} = I_{3,1}$, or $I_{2,3} = I_{3,2}$.

True.

A principal moment of inertia is a diagonal entry in the tensor: $I_{1,1}$, $I_{2,2}$, or $I_{3,3}$. This is the semantic of moment of inertia discussed in elementary treatment of Physics.

This is not true, the principal moments of inertia are the diagonal elements (eigenvalues), only if you have diagonalized the inertia matrix (which you can always do).

What is the term for $I_2$ and $I_3$ in the last line below?
   $$I_{1,1} = I_2 + I_3 $$

If $ I_{2}=I_{2,2}$ and  $ I_{3}=I_{3,3}$, once you have diagonalized $I$. In general, we have (dropping the indices):
$$I = m \begin{bmatrix}y^2+z^2 & I_{1,2} & I_{1,3} \\ I_{2,1} & x^2+z^2  & I_{2,3} \\ I_{3,1} & I_{3,2} & x^2+y^2\\\end{bmatrix}$$ 
But in a planar body one of the coordinates is zero, in the example below (continuos body but same concept), $y=0$. Then you can see that:
$$I = \int d\mathbf{m} \begin{bmatrix}z^2 & I_{1,2} & I_{1,3} \\ I_{2,1} & x^2+z^2  & I_{2,3} \\ I_{3,1} & I_{3,2} & x^2\\\end{bmatrix}$$ 
So the moment of inertia associated to the z axis is the sum of the other two. 

A: They are just the equation above rewritten to a shorter format using the definition of the moment of inertia about an axis:
$$
I = \sum_{i=1}^N{m_ir_i^2}.
$$
The equation above uses $r_{j,2}$ and $r_{j,3}$, which correspond to $I_2$ and $I_3$.
