# Impose Uniqueness on Diagonalization of Inertia Tensor?

## Main Question or Discussion Point

Given an inertia tensor of a rigid body I, one can always find a rotation that diagonalizes I as I = RT I0 R (let's say none of the value of the inertia in I0 equal each other, though). R is not unique, however, as one can always rotate 180 degrees about a principal axis, or rearrange the entries of I0 via rotations. I'm curious if there a set of conditions that one can impose on R to make it unique, however. One can eliminate the ordering issue by insisting that the entries of I0 are in sorted order. What about the 180 degree rotation issue, is there an additional condition that one can impose to eliminate this ambiguity?

## Answers and Replies

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vanhees71
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2019 Award
You can always diagonalize a symmetric matrix with a orthogonal transformation. The direction of the principal axes is unique if no two eigenvalues are the same. So the only freedom you have after ordering the eigenvalues in the diagonal for $R$ is the direction of the axes. As an additional constraint you can only impose the condition that the transformation matrix is not only orthogonal but even a rotation, i.e., an SO(3) matrix (with determinant +1). Then still you have the freedom to switch any two of the chosen eigenvectors.

So I don't think that you can make the transformation matrix unique, but why should you want this anyway? It's good enough to have one body-fixed basis where the inertia tensor is diagonal.

Andy Resnick