Ellipsoid

A symmetric and positive definite matrix $$A=A^T \in \mathbb{R}^{n\times n} > 0$$ defines an ellipsoid $$\mathcal E$$ through a quadratic form as :

$$\mathcal E = \{x : x^T A x \le 1 \}$$

In this case the ellipsoid is centered on zero. The ellipsoid is illustrated for the 2D case below:



The semi-axes are given by $$s_i = \lambda_i^{-1/2} q_i$$. That is, eigenvectors $$q_i$$determine the directions and eigenvalues $$ \lambda_i$$ determine the lengths of semi-axes. $$\sqrt{\lambda_{\text{max}}/\lambda_{\text{min}}}$$ gives the maximum eccentricity of the ellipsoid.

Interpretation
The ellipsoids provide an intuitive description of the conditions under which two symmetric and positive definite matrices are comparable. The ellipsoids of two symmetric and positive definite matrices $$A,B$$ can be used to compare the two matrices. That is, suppose that $$\mathcal{\tilde{E}} := \{x : x^T B x \le 1\}$$, and $$\mathcal E := \{ x: x^T A x \le 1 \} $$. Then,

$$A\ge B \iff \mathcal E \subseteq \mathcal{\tilde{E}}$$