Conditioning

How much does the answer wobble when the data wobbles? That sensitivity is conditioning, and for a linear problem it is captured by a single number built from the singular values — the condition number

\kappa(G) = \frac{\sigma_{\max}}{\sigma_{\min}}.

It bounds how a relative error in the data becomes a relative error in the solution:

\frac{\|\delta m\|}{\|m\|} \le \kappa(G)\,\frac{\|\delta d\|}{\|d\|}.

A small \kappa (near 1) is well-conditioned: errors stay the same size. A huge \kappa is ill-conditioned: a 0.1% data error can become a 100% solution error.

The ellipse aspect ratio

Picture the forward map sending the unit circle to an ellipse: its longest semi-axis is \sigma_{\max}, its shortest \sigma_{\min}, and \kappa is the aspect ratio. A near-circular ellipse (\kappa \approx 1) is easy to invert; a long thin sliver (\kappa \gg 1) squashes one direction almost to nothing, and undoing that squash magnifies any error along it. Conditioning is a property of the operator, set before any noise enters — it tells you how much trouble to expect.

\kappa(G) = \sigma_{\max}/\sigma_{\min}: the ellipse aspect ratio.
It bounds error amplification: \|\delta m\|/\|m\| \le \kappa\,\|\delta d\|/\|d\|.
\kappa \approx 1 well-conditioned; \kappa \gg 1 ill-conditioned.