Numerical Stability

Numerical methods can fail silently when matrices are ill-conditioned. The condition number $\kappa(A) = \sigma_1 / \sigma_n$ (ratio of largest to smallest singular values) measures sensitivity. A high $\kappa$ means small changes in input produce huge changes in output.

For linear systems $Ax = b$, the relative error in $x$ can be up to $\kappa(A)$ times the relative error in $b$. With double precision ($\sim 16$ digits), $\kappa = 10^{10}$ leaves about $6$ trustworthy digits.

Common pitfalls:

Computing $X^\top X$ before solving regression squares the condition number. Use QR or SVD directly.Inverting matrices is a code smell. Solve linear systems instead — same answer, more stable.Subtracting nearly-equal numbers loses precision (catastrophic cancellation).

In quant work, near-singular covariance matrices appear all the time (e.g. when assets are highly correlated). Regularization (shrinkage, ridge) and PCA-based dimensionality reduction are the standard defenses.