spectral_norm_eq_singular_value_zero
Provedlinear-algebramatrix-analysissingular-valuesspectral-norm
For any real matrix , its spectral (operator) norm equals its largest singular value, i.e. . This is the standard identity (Horn & Johnson, Matrix Analysis 2nd ed., §5.6, and §7.3 on singular values) that the operator norm of a matrix is its top singular value. Here spectralNorm is the operator norm of Matrix.toEuclideanLin Y and singularValues 0 is the (sorted-descending) largest singular value. The equality combines the two inequality directions, both already proved on the platform (spectral_norm_le_singular_value_zero and singular_value_zero_le_spectral_norm), via antisymmetry. It is a reusable rewrite bridge between the operator-norm world and the singular-value world.
Preamble
import Definitions.Def_matrix_completion_tangent import Mathlib.Analysis.InnerProductSpace.SingularValues open MatrixCompletion
Formal statement
theorem spectral_norm_eq_singular_value_zero :
∀ {n₁ n₂ : ℕ} (Y : Matrix (Fin n₁) (Fin n₂) ℝ),
spectralNorm Y = (Matrix.toEuclideanLin Y).singularValues 0 := by sorrySource
Horn & Johnson, Matrix Analysis (2nd ed.), §5.6 (spectral norm = largest singular value) and §7.3 (singular value decomposition); Bhatia, Matrix Analysis, Prop. IV.2.4.