For rotation about an arbitrary axis, a rigid body's resistance to angular acceleration cannot generally be described by a single scalar moment of inertia. The relation between angular velocity $\boldsymbol\omega$ and angular momentum $\mathbf L$ is $$\mathbf L=\mathbf I\boldsymbol\omega,$$ where $\mathbf I$ is the inertia tensor.

For point masses, $$I_{ij}=\sum_a m_a\left(r_a^2\delta_{ij}-x_{a,i}x_{a,j}\right).$$ For a continuous body the sum becomes a volume integral. The diagonal terms are moments of inertia about coordinate axes; off-diagonal terms encode products of inertia and coupling between axes.

Because $\mathbf I$ is real and symmetric, it has orthogonal eigenvectors. These are the body's principal axes, and the eigenvalues $I_1,I_2,I_3$ are the principal moments of inertia. In principal-axis coordinates, $$\mathbf I=\begin{pmatrix}I_1&0&0\0&I_2&0\0&0&I_3\end{pmatrix},$$ so $$L_i=I_i\omega_i.$$ Only when rotation is exactly about a principal axis are $\mathbf L$ and $\boldsymbol\omega$ guaranteed to be parallel.

For a thin rectangular plate centered at the origin with sides aligned to its symmetry axes, symmetry makes the off-diagonal products of inertia vanish. Those symmetry axes are therefore principal axes automatically.

Diagonalizing the inertia tensor turns an awkward three-dimensional rotation problem into its natural coordinates. The same eigenvector idea used throughout linear algebra now acquires a direct mechanical meaning: principal axes are directions in which rotational inertia acts independently.