Home/Algebra/Matrix norms and conditioning/Spectral norm24. Matrix norms and conditioningSpectral normLevel: avanzadoIn 24. Matrix norms and conditioningFormulaCopy LaTeX∥A∥2=σmax(A).\|A\|_2 = \sigma_{\max}(A).∥A∥2=σmax(A).AlsoCopy LaTeX∥A∥2=λmax(ATA)\|A\|_2 = \sqrt{\lambda_{\max}(A^TA)}∥A∥2=λmax(ATA)Tagsalgebraalgebra formulalinearnormsnorms conditioningDetailIt is the largest Euclidean stretching factor produced by a matrix.Explore the ideaFound an error? Contact us.Related formulasRelatedInduced matrix norm∥A∥p=maxx≠0∥Ax∥p∥x∥p.\|A\|_p = \max_{\mathbf x\neq\mathbf0} \frac{\|A\mathbf x\|_p}{\|\mathbf x\|_p}.∥A∥p=x=0max∥x∥p∥Ax∥p.Open formulaOpen in sectionRelatedSVDA=UΣVTA=U\Sigma V^TA=UΣVTOpen formulaOpen in sectionRelatedEigenvalue equationAv=λv,v≠0Av=\lambda v,\quad v\ne0Av=λv,v=0Open formulaOpen in sectionRelatedCondition numberκ(A)=∥A∥ ∥A−1∥.\kappa(A) = \|A\|\,\|A^{-1}\|.κ(A)=∥A∥∥A−1∥.Open formulaOpen in section