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## 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?

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