24. Matrix norms and conditioning
Condition number
Level: advanced
For an invertible matrix and a compatible norm
In the 2-norm
Detail
Measures the sensitivity of a linear problem to small perturbations in the data.
Explore the idea
You'll see that the condition number tells you if a system is sensitive to errors.
Try it — Make (low ): the mesh flattens and the problem becomes ill-conditioned.
Aproximación de bajo rango
Se truncan las componentes singulares pequeñas: A_k = Σ_{i≤k} σ_i u_i v_iᵀ conserva la energía dominante y descarta el resto.
Pruébalo — Mueve k y observa barras, A_k y el residual. En Compresión compara almacenamiento mn vs k(m+n+1).
Eckart–Young: A_k es óptima en ‖·‖F y ‖·‖₂ entre las matrices de rango ≤ k.
k2
rango exacto = 4k = 2energía 97.4%‖A−Aₖ‖F = 1.54‖A−Aₖ‖₂ ≈ σ3 = 1.5
Valores singulares σᵢ
8.4σ1
4.1σ2
1.5σ3
0.35σ4
conservados (i≤k) descartados
Ak = σ1 u1 v1ᵀ + σ2 u2 v2ᵀ
A
7.91.1-1.10.80.640.6-0.1-0.70.61.60.30.5-0.60.10.51.50.4-0.60.2
Aₖ
7.91.2-1.30.80.63.90.9-0.03-0.90.90.4-0.10.5-0.6-0.20.11.60.3-0.20.2
A − Aₖ
0.04-0.10.20.02-0.10.1-0.3-0.030.2-0.31.20.40.02-0.10.30.4-0.10.1-0.30.01
k=2/4 · energía 97.4% · ‖A−Aₖ‖F=1.54
Conditions
- The formula \|A\|\|A^{-1}\| requires A to be invertible.
- For a singular matrix, the condition number is considered infinite in this context.