Find rotation matrix that diagonalizes given inertia tensor

[mindarson]

Homework Statement

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.

Homework Equations

I' = RIR^-1

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?

 

[SteamKing]

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]

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]

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