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The moment of inertia tensor from physics looks very similar to the covariance matrix, used for PCA. How are their eigenvectors and eigenvalues related?

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First, let's set up the moment of inertia tensor for $N$ points, where point $n$ has mass $m_n$ and coordinates $(x_m^{(1)}, x_m^{(2)}, ...)$ and define $C_{ij} = \sum_{n=1}^N m_n x_n^{(i)} x_n^{(j)} $. Then the moment of inertia tensor is

$\mathbf{J} = tr(C)\mathbf{I}-\mathbf{C}$

The symbol $\mathbf{I}$ denotes the identity matrix, and $\mathbf{J}$ is the moment of inertia tensor. If $\mathbf{v}$ is an eigenvector of $\mathbf{C}$ with eigenvalue $\lambda$ then $\mathbf{C} \lambda = \lambda \mathbf{v}$. Also, we have $tr(C)\mathbf{v} = tr(C)\mathbf{I} \mathbf{v}$. Subtracting these two equations gives:

$(tr(C)-\lambda)\mathbf{v} = ( tr(C)\mathbf{I}-\mathbf{C})\mathbf{v}$

so $tr(C)-\lambda$ is an eigenvalue of $\mathbf{J}$ and $\mathbf{v}$ is an eigenvector of $\mathbf{J}$. And finally, the weighted covariance matrix is $\mathbf{K}=\mathbf{C}/M$ where $M$ is the total mass and the covariance weights correspond to the masses.

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