Conditioning of Matrix I

Geometry of the image ellipse

Formulation

\kappa(\mathbf{x})_{\mathbf{x} \mapsto A\mathbf{x}} = \frac{||A|| \cdot ||\mathbf{x}||}{||A\mathbf{x}||}

\le \kappa(A) = ||A|| \cdot ||A^{-1}||

The unit circle in the domain $\mathbb{R}^2$ maps to an ellipse lying inside the plane Span(A) in $\mathbb{R}^3$ .

The worst-case perturbation $A\delta\mathbf{x}$ always aligns with the long axis of the ellipse.
When $A\mathbf{x}$ is near the short axis, $||A\mathbf{x}||$ is small, so $\kappa(\mathbf{x})$ is large.
Pointwise conditioning depends on both the input location $\mathbf{x}$ and the worst-case perturbation amplification.

φ (phi)45°

Position of x on the unit circle in the domain.

Domain (R²)

x = (0.71, 0.71)

||Ax||

1.458

κ(x)

1.372

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