The general moment of inertia tensor is represented by a real-symmetric matrix. Show that there exist an orthogonal matrix and diagonal matrix such that
Furthermore, deduce that
We first prove a general property in spectral theory: If is a Hermitian matrix, there exists a unitary matrix , the conjugate transpose of the unitary matrix and a diagonal matrix such that .
We begin by finding such that , where is an upper triangular matrix. Since Hermitian matrices follow the property , we can deduce the following:
The conjugate transpose of an upper triangular matrix is a lower triangular matrix. Let be the lower triangular matrix. From the observation that implies .
However, we know that the inertia tensor is real-symmetric. Hence, it follows the property: If is a real-symmetric matrix, there exists an orthogonal matrix , the transpose of the orthogonal matrix and a diagonal matrix such that .
Since real-symmetric matrices satisfy the property and orthogonal matrices satisfy the property , the proof is structurally identical to the one above.
Therefore, it follows that
where is the rotation matrix, and is a diagonal matrix containing the principle axes of rotation.
Check out my other notes at Proof, Disproof, and Derivation