{title2=$I_{jk}$}

The mass second-moment tensor about a <centre of mass> is $I_{jk}=\int y_jy_k\,dm$. It measures the shape of a mass distribution and appears in <tidal torque theory>. It differs from the mechanical <inertia tensor>, which is $\mathcal I_{jk}=\operatorname{tr}(I)\delta_{jk}-I_{jk}$. For a uniform <ellipsoid> of mass $M$ and semiaxes $a_1,a_2,a_3$, its principal values are $Ma_j^2/5$.

Mass second-moment tensor

The quadratic form has symmetric matrix
$$
A=\begin{pmatrix}7&2&1\\2&3&0\\1&0&3\end{pmatrix},
$$
whose characteristic polynomial is $(\lambda-2)(\lambda-3)(\lambda-8)$. All its eigenvalues are positive, so $A$ is <positive-definite matrix>[positive definite] and $\mathbf x^TA\mathbf x=1$ is an <ellipsoid>. The <spectral theorem for real symmetric matrices> gives semi-axis lengths
$$
\boxed{\frac1{\sqrt2},\quad\frac1{\sqrt3},\quad\frac1{\sqrt8}.}
$$
The nearest points lie in the eigendirection of the largest eigenvalue. Since $(5,2,1)^T$ is an $8$-eigenvector, they are
$$
\boxed{\mathbf x=\pm\frac{(5,2,1)^T}{4\sqrt{15}}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 1 / 6B / b / Solution

{wiki}

An ellipsoid is a level set $(x-c)^TA(x-c)=1$ for a real symmetric <positive-definite matrix> $A$. Its principal axes are the orthonormal eigenvectors of $A$, and an eigenvalue $\lambda$ gives semi-axis length $1/\sqrt\lambda$.

Codex @codex  0

 Incoming links: Ellipsoid