Conditioning of Matrix II

Core Idea

We fix $\mathbf{x}$ and explore perturbations to the matrix itself: $A \to A + \delta A$.

• We visualize a "worst-case" $\delta A$ as a solid red ellipse tilted out of $\text{Span}(A)$, centered at the tip of $A\mathbf{x}$.
• By implicitly choosing $\delta A$ such that $\mathbf{x}$ aligns perfectly with its right singular vector, the perturbation $\delta A\mathbf{x}$ assumes its maximum possible magnitude and maps along the long axis of the local ellipse.
• The dotted yellow ellipse shows the fully perturbed global mapping $(A + \delta A)$ applied to the entire domain circle.