Moment of inertia tensor calculation and diagonalization

[BiGyElLoWhAt]

Homework Statement

Not sure if this is advanced, so move it wherever.
A certain rigid body may be represented by three point masses:
m_1 = 1 at (1,-1,-2)
m_2 = 2 at (-1,1,0)
m_3 = 1 at (1,1,-2)

a) find the moment of inertia tensor
b) diagonalize the matrix obtaining the eigenvalues and the principal axes (as orthogonal vectors)

Homework Equations

$I_{ij} = m_{\beta}(\delta_{ij}r_{\beta}^2 - x_{i\beta}x_{j\beta})$
$\vec{A} = \vec{P^{-1}}\vec{D}\vec{P}$
$I_{ij}=I_{ji}$

The Attempt at a Solution

I'm going to drop the beta's, but each xyz is associated with the mass attached to the term.
$I_{00} = m_1(x_0^2 + x_1^2 +x_2^2 - x_0^2) + m_2(...) + m_3(...)$
$= 1(5) + 2(1) + 1(5) = 12$
$I_{01} = m_1(-x_0x_1) + m_2(...) +m_3(...)$
$= -1(-1) -2(-1) -1(1) = 2$
$I_{02} = m_1(-x_0x_2) + m_2(...) + m_3(...)$
$= -1(-2) -2(0) -1(1) = 4$
$I_{11} = m_1(x_0^2+x_2^2) + m_2(...) + m_3(...)$
$= 1(5) +2(1) +1(5) = 12$
$I_{12} = m_1(-x_1x_2) + m_2(...) +m_3(...)$
$= -1(2) +2(0) - 1(-2) = 0$
$I_{22} = m_1(x_0^2 +x_1^2) + m_2(...) +m_3(...)$
$=1(2) +2(2) +1(2) = 8$
This gives me
$\vec{I} = \left ( \begin{array}{ccc} 12 & 2 & 4 \\ 2 & 12 & 0 \\ 4 & 0 & 8 \\ \end{array} \right )$

and

$det(I_{\lambda}) = 0 = -\lambda^3 + 32\lambda^2 - 316\lambda + 928$
I don't even know how to solve that equation, and online calculators give
diagonal(5.35139, 11.2841, 15.3627)
Did I mess up? There's nothing about using a calculator, but I don't know how else to solve this. Did I mess up? I am pretty sure I'm not supposed to end up with a bunch of stupid numbers like this.

 

[vela]

For what it's worth, I got the same matrix you did.

 

[BiGyElLoWhAt]

Well, maybe I'll just roll with it then. I thought I checked all my algebra carefully. I'm pretty sure I'm supposed to use P^-1 D P to get the diagonal since he said "obtaining the eigenvectors", but maybe I'll just do row operations or something to avoid the decimals. Thanks.

 

[mpresic]

Not sure but it looks as if you are using vectors with respect to the origin. Shoudn't you be using vectors relative to the center of mass of the system

 

[mpresic]

instead (for mass 1) of x0 = 1 x1 = -1 and x2 = -2 in line 1 of your calculation: try x0 = 1 - x0cm = 1 - 0 = 1 x1 = -1 = x1 cm = -1 - 1/2 = -3/2 and x2 = -2 - x2 cm = -2 - -1 , etc.

this means a lot more calculation because you need to subtract off the center of masses but I think it will lead to a better answer. On the other hand, I'm glad I am not doing the calculation.

 

[mpresic]

I ended up with eigenvalues 11; 11/2 +/- sqrt(57) / 2.

 

[BiGyElLoWhAt]

That is a lot neater. I didn't think about doing that. I will keep it in mind for next time.