Conditioning of LSE II

Under Construction

Sensitivity to matrix perturbations $\delta A$

Formulation

A\mathbf{x} \approx \mathbf{b}

\mathbf{\hat{y}} = A(A^*A)^{-1}A^*\mathbf{b}

\kappa_{A \mapsto \mathbf{\hat{y}}} \sim \frac{\|\mathbf{r}\|}{\cos \theta}

We fix $\mathbf{b}$ along the minor axis $\sigma_n$ of $A$ to visualize the worst-case relative error.

No matter how the column space is tilted, the line $\mathbf{b} - \mathbf{\hat{y}}'$ remains at a right angle to $0 - \mathbf{\hat{y}}'$ .
Therefore, as $\text{Span}(A)$ is tilted towards $\mathbf{b}$ , $\mathbf{\hat{y}}'$ moves strictly along a sphere of radius $\|\mathbf{b}\|/2$ centered at $\mathbf{b}/2$ .

θTilt Angle45°

σ₂(δA)

0.40

0 (Rank-1)0.5 (Full)

||ŷ||

1.000

||r||

1.000

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