Editing Moment of inertia (section)

=== Angular momentum ===
Note that the [[Cross product#Conversion to matrix multiplication|cross product can be equivalently written as matrix multiplication]] by combining the first operand and the operator into a skew-symmetric matrix, <math>\left[\mathbf{b}\right]</math>, constructed from the components of <math>\mathbf{b} = (b_x, b_y, b_z)</math>:
<math display="block">\begin{align}
  \mathbf{b} \times \mathbf{y}
                           &\equiv \left[\mathbf{b}\right] \mathbf{y} \\
  \left[\mathbf{b}\right] &\equiv \begin{bmatrix}
                 0   & -b_z &  b_y \\
                 b_z &    0 & -b_x \\
                -b_y &  b_x &    0
              \end{bmatrix}.
\end{align}</math>

The inertia matrix is constructed by considering the angular momentum, with the reference point <math>\mathbf{R}</math> of the body chosen to be the center of mass <math>\mathbf{C}</math>:<ref name="Marion 1995"/><ref name="Kane"/>
<math display="block">\begin{align}
  \mathbf{L}
    &= \sum_{i=1}^n m_i\,\Delta\mathbf{r}_i \times \mathbf{v}_i \\
    &= \sum_{i=1}^n m_i\,\Delta\mathbf{r}_i \times \left(\boldsymbol{\omega} \times \Delta\mathbf{r}_i + \mathbf{V}_\mathbf{R}\right) \\
    &= \left(-\sum_{i=1}^n m_i\,\Delta\mathbf{r}_i \times \left(\Delta\mathbf{r}_i \times \boldsymbol{\omega}\right)\right) + \left(\sum_{i=1}^n m_i \,\Delta\mathbf{r}_i \times \mathbf{V}_\mathbf{R}\right),
\end{align}</math>
where the terms containing <math>\mathbf{V_R}</math> (<math>= \mathbf{C}</math>) sum to zero by the definition of [[center of mass]].

Then, the skew-symmetric matrix <math>[\Delta\mathbf{r}_i]</math> obtained from the relative position vector <math>\Delta\mathbf{r}_i = \mathbf{r}_i - \mathbf{C}</math>, can be used to define,
<math display="block">
  \mathbf{L} =
  \left(-\sum_{i=1}^n m_i \left[\Delta\mathbf{r}_i\right]^2\right)\boldsymbol{\omega} =
  \mathbf{I}_\mathbf{C} \boldsymbol{\omega},
</math>
where <math>\mathbf{I_C}</math> defined by
<math display="block">\mathbf{I}_\mathbf{C} = -\sum_{i=1}^n m_i \left[\Delta\mathbf{r}_i\right]^2,</math>
is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass <math>\mathbf{C}</math>.