A ball on $\R^n$ centered at the origin can be defined in the following way,
$$ \mathcal{B} = \{ x \in \R^n \mid x = rI_n y, \space \|y\|_2 \leq 1\} $$
This notion can be generalized to the ellipsoid centered at the origin,
$$ \mathcal{Q} = \{x \in \R^n \mid x = By, \space ||y||_2 \leq 1\} = \{x \in \R^n \mid x^T Px \leq 1, \text{positive-definite $P$}\} $$
Finally, this can be generalized to the case where $B$ is a general $m \times n$ matrix,
$$ \mathcal{E} = \{x \in \R^n \mid x = By , ||y || \leq 1\} $$
Note that $\mathcal{E}$ is just the image of the unit ball
<aside> 💡
Left Singular Vectors of $B$ are semi-axes. For an ellipsoid $\mathcal{E}$ defined by $B \in \R^{n \times m}$ with rank $r$, the semi-axes of $\mathcal{E}$ are the left singular vectors of $B$, with the semi-axis associated with $u_i$ having length $\sigma_i$.
</aside>
Perturbing a measurement $y \in \R^{m}$ by at most a unit amount (i.e. $||\Delta y|| \leq 1$) leads to an interesting geometry of perturbations of the solution set.
$$ \min _{x +\Delta x \in \R^n} ||A(x+\Delta x)-(y+\Delta y)||_2 \implies S = x^* +\Delta x^* +\text{null } A \\ x= A^{\dagger}y \hspace{15pt} \Delta x = A^\dagger \Delta y $$
The object of study is the set of these solution perturbations $\mathcal{E}$.
$$ \mathcal{E} := \{\Delta x \in \R^n \mid \Delta x = A^\dagger \Delta y , \space ||\Delta y|| \leq 1\} $$
By SVD, $A = U \Sigma V^T$, so $A^\dagger = V \Sigma ^{\dagger} U^T$ and thus has left singular vectors $v_1, \dots, v_n$ with singular values $\frac{1}{\sigma_1}, \dots, \frac{1}{\sigma_r}, 0, \dots, 0$. Since $\mathcal{E}$ takes the form of an ellipsoid, these objects fully characterize the geometry of the solution perturbation set.