equation of major axis of an ellipsoid

Question

What is the equation of 3 major axes of the following ellipsoid? $$ \begin{pmatrix}x & y & z\end{pmatrix} \begin{pmatrix} \alpha_1 & \beta_3 & \beta_2\\ \beta_3 & \alpha_2 & \beta_1\\ \beta_2 & \beta_1 & \alpha_3 \end{pmatrix} \begin{pmatrix}x\\y\\z\end{pmatrix} = 1\tag{1}$$

Answer 1

Your equation can be rewritten as $$ \mathbf{x}^\top\Sigma\mathbf{x}=1, $$ where $\mathbf{x}=(x, y, z)^\top$, and $$ \Sigma = \begin{pmatrix} \alpha_1 & \beta_3 & \beta_2\\ \beta_3 & \alpha_2 & \beta_1\\ \beta_2 & \beta_1 & \alpha_3 \end{pmatrix}. $$ If you take the SVD (Singular Value Decomposition) of $\Sigma$ you will have $$ \Sigma = VDV^\top, $$ where $D$ is the diagonal matrix containing the eigenvalues of $\Sigma$, and $V$ is an orthonormal matrix, which contains the axes (its columns). Say, $$ D=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3). $$ The lengths of the axes are given as $$ a = \frac{1}{\sqrt{\lambda_1}}, b = \frac{1}{\sqrt{\lambda_2}}, c = \frac{1}{\sqrt{\lambda_3}}. $$ Then, you can have your ellipsoid in the following form: $$ \Big(\frac{x}{a}\Big)^2 + \Big(\frac{y}{b}\Big)^2 + \Big(\frac{z}{c}\Big)^2 = 1. $$

EDIT: If you need just the axes, then you can find them in the columns of $V$.