Condition Number

Condition Number

The condition number $\kappa(A)$ measures how sensitive a linear system $Ax = b$ is to small perturbations:

$\kappa(A) = \frac{\lambda_{\max}}{\lambda_{\min}} \quad (\text{for symmetric positive definite matrices})$

A large condition number means the system is ill-conditioned: tiny errors in $b$ can cause enormous errors in $x$.

2×2 Eigenvalues via Trace and Determinant

For a 2×2 symmetric matrix, eigenvalues are:

$\lambda = \frac{\text{tr} \pm \sqrt{\text{tr}^2 - 4 \cdot \det}}{2}$

where $\text{tr} = A_{00} + A_{11}$ and $\det = A_{00} A_{11} - A_{01} A_{10}$.

Interpretation

Examples

$A = \begin{pmatrix}3&1\\1&3\end{pmatrix}: \; \text{tr}=6,\ \det=8,\ \lambda = \tfrac{6\pm2}{2} = 4, 2 \;\ \Rightarrow \;\ \kappa = 2.0$

$B = \begin{pmatrix}2&1\\1&2\end{pmatrix}: \; \text{tr}=4,\ \det=3,\ \lambda = 3, 1 \;\ \Rightarrow \;\ \kappa = 3.0$

$C = I: \; \lambda = 1, 1 \;\ \Rightarrow \;\ \kappa = 1.0$

Your Task

Implement condition_number_2x2(A) that computes $\kappa = \lambda_{\max} / \lambda_{\min}$ for a 2×2 symmetric positive definite matrix using the trace-determinant formula.