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# 13.5: Matrix and Tensor Formulations of Rigid-Body Rotation

The prior notation is clumsy and can be streamlined by use of matrix methods. Write the inertia tensor in a matrix form as

$\{\mathbb{I}\}= \begin{pmatrix} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \end{pmatrix}$

The angular velocity and angular momentum both can be written as a column vectors, that is

$\boldsymbol{\omega}=\begin{pmatrix} \omega_{1} \\ \omega_{2} \\ \omega_{3} \end{pmatrix} \quad \mathbf{L}=\begin{pmatrix} L_{1} \\ L_{2} \\ L_{3} \end{pmatrix}$

As discussed in appendix $$19.5.2$$, Equation $$13.4.7$$ now can be written in tensor notation as an inner product of the form

$L = \{\mathbb{I}\} \cdot \boldsymbol{\omega}$

Note that the above notation uses boldface for the inertia tensor $$\mathbb{I}$$, implying a rank-2 tensor representation, while the angular velocity $$\boldsymbol{\omega}$$ and the angular momentum $$\mathbf{L}$$ are written as column vectors. The inertia tensor is a 9-component rank-2 tensor defined as the ratio of the angular momentum vector $$\mathbf{L}$$ and the angular velocity $$\boldsymbol{\omega}$$.

$\{\mathbb{I}\} = \frac{\mathbf{L}}{\boldsymbol{\omega}}$

Note that, as described in appendix $$19.5$$, the inner product of a vector $$\boldsymbol{\omega}$$, which is the rank 1 tensor, and a rank 2 tensor $$\{\mathbb{I}\}$$, leads to the vector $$\mathbf{L}$$. This compact notation exploits the fact that the matrix and tensor representation are completely equivalent, and are ideally suited to the description of rigid-body rotation.