Help With Tensors: Solving a Problem

[atomicpedals]

I'm very clearly not understanding something, if someone could help me put my finger on what that something is. So here's what I've got:

Problem: Three equall mass points (mass m) are located at (a,a) (a,‐a) and (‐a,‐a) (all have z=0).

a) Show I = ma^2 (3ii − ij − ji + 3jj + 6kk)

I11 I12 I13
I = ma^2 I21 I22 I23
I31 I32 I33

I11 I12 0
I = ma^2 I21 I22 0
0 0 I33

3 −1 0
I= ma^2 −1 3 0
0 0 6

I = ma2 (3ii − ij − ji + 3jj + 6kk)

b) Find the principle set of axes u1, u2, and u3 in terms of I, j, and k.
Note: I’m totally sure what exactly to go for...

c) Show
I=ma2(4u1u1 +2u2u2 +6u3u3)
= ma2 (3ii − ij − ji + 3jj + 6kk)
= ma2 [Aii + Bjj + Ckk]
= ma2[4ii + 2jj+ 6kk]
=ma2(4u1u1 +2u2u2 +6u3u3)

d) Show by explicit calculation that the products of inertia with respect to the new
axes are zero.

This solution follows from part b, once I get part b hammered out this will then follow.

 

[Mindscrape]

For part b) you want to diagonalize your matrix, i.e. find the eigenvalues. Since inertia tensors are symmetric they are diagonalizable.