Find rotation matrix that diagonalizes given inertia tensor

mindarson

1. The problem statement, all variables and given/known data

In a particular coordinate frame, the moment of inertia tensor of a rigid body is given by

I = {{3,40},{4,9,0},{0,0,12}}

in some units. The instantaneous angular velocity is given by ω = (2,3,4) in some units. Find a rotation matrix a that transforms to a new coordinate system in which the inertia tensor becomes diagonal.

2. Relevant equations

I' = RIR^-1

3. The attempt at a solution

I've tried a couple of approaches. First, I figure that I'm essentially just trying to diagonalize a matrix (the problem does not specify a particular transformation, just one that will diagonalize the tensor). So I used the equation above and found the diagonalizing matrix R by finding the eigenvectors of I and using them as the columns of the diagonalizing matrix R. I do not see any reason why this will not work, but when I checked it the matrix I came up with for R did not diagonalize I. But maybe my computations were hasty.

My second approach was to assume a certain transformation, i.e. a rotation of the body about, say, the x3 axis. Then the rotation matrix would be

R = {{cosθ, sinθ, 0},{-sinθ,cosθ,0},{0,0,1}}

and its inverse is

R^-1 ={{cosθ,-sinθ,0},{sinθ,cosθ,0},{0,0,1}}

I then compute the matrix I' = RIR^-1 and try to find a rotation angle θ that causes the off-diagonal elements to vanish. However, I have no idea how to proceed to find this angle. (Solve another system of equations?) Also, I have a distinct feeling that the solution should not be quite this complicated.

Can anyone give guidance?

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SteamKing

Staff Emeritus
Homework Helper
Your initial approach should have done the trick.

mindarson

Thank you for your reply! I tried my initial approach again, and this time it did work to diagonalize the tensor. I think I just was hasty in my calculations.

However, I wonder if you can help me think this through. I do not feel too comfortable just calling the matrix I found via the eigensystem a 'rotation matrix' without actually dealing explicitly with the angles and functions of the angles through which the coordinate system has been rotated. Is there a way, at least in principle, to arrive at this rotation matrix using my second approach, i.e. by using the transformation matrix that uses the directional cosines? Then I would feel more comfortable with my answer from a physical, versus just a formal, perspective.

But thanks again for your confirmation!

SteamKing

Staff Emeritus
Homework Helper
Both approaches are identical. One property of a rotation matrix R which you might have overlooked is that R Transpose = R inverse.

See wiki articles on Mass Moment ofIinertia and Rotation Matrix for the math.

vela

Staff Emeritus
Homework Helper
Just normalize your eigenvectors. You should be able to see by inspection what $\cos\theta$ and $\sin\theta$ are equal to.

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