the matrix condition number
When the problem is 'solve A x = b' — find the x that the matrix A maps to the vector b — we need a condition number for the whole linear system. The matrix condition number kappa(A) tells you how much solving such a system can amplify errors in the data b (or in A itself). A small kappa(A) means a stable, trustworthy system; a huge kappa(A) means the matrix is close to being singular and the solution is treacherous.
The definition is kappa(A) = ||A|| times ||A^{-1}||, using a matrix norm. The geometric meaning is cleanest with the 2-norm: kappa(A) equals the largest singular value of A divided by the smallest, sigma_max / sigma_min. Picture A stretching the unit sphere into an ellipsoid; kappa(A) is the ratio of its longest to shortest axis — how lopsided the stretching is. A kappa of 1 means A acts like a rotation or uniform scaling (perfectly conditioned); a near-zero smallest singular value sends kappa toward infinity, meaning A nearly collapses some direction to nothing and can barely be undone.
The practical payoff is a bound: the relative error in the computed solution x is at most about kappa(A) times the relative error in b. So if kappa(A) is about 10^8, expect to lose roughly 8 of your 16 double-precision digits even with a flawless solver. This is why numerical libraries return an estimate of kappa(A) (LAPACK's RCOND) alongside the solution: it is your warning label. Note kappa(A) is always at least 1 — no matrix is better than perfectly conditioned.
The 5-by-5 Hilbert matrix (entries 1/(i+j-1)) has kappa(A) about 4.8 x 10^5; the 10-by-10 has kappa about 1.6 x 10^13 — solving Hilbert systems in double precision is hopeless past size ~12 because the smallest singular value all but vanishes.
A textbook ill-conditioned matrix: small, innocent-looking, and numerically poisonous.
kappa(A) depends on the chosen norm, but all common choices agree on whether a matrix is well- or ill-conditioned. It measures the conditioning of solving A x = b, not the difficulty of factoring A.