The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product

Depends on

• Real and complex inner product spaces, with the inner product linear in the first argument
• Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$

Used by

• Certified-action implementation of exponential Euler and ETD2 Algorithm
• Finite scaled Taylor action for the matrix exponential Algorithm
• Flexible GMRES with variable right preconditioning Algorithm
• Rank-thresholded least squares by the singular value decomposition Algorithm
• The induced length is a norm Corollary
• The inner-product norm is definite, homogeneous, and satisfies the triangle inequality Corollary
• A conjugate point at which there are many geodesics Counterexample
• Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group Definition
• Covariance matrix of a finite-dimensional random vector Definition
• Finite block encoding Definition
• Householder reflectors in real or complex inner-product spaces Definition
• Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces Definition
• Normwise certified matrix-vector and vector-step arithmetic Definition
• Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases Definition
• Rayleigh-quotient iteration for Hermitian matrices Definition
• Real and complex inner-product spaces and their induced length Definition
• Real Givens rotations and complex Givens transformations Definition
• The Arnoldi process for building an orthonormal basis of a Krylov subspace Definition
• The B-inner product, B-norm, and dual residual norm Definition
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
• The normwise condition number of a simple eigenvalue Definition
• The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖ Definition
• The phi-function family by everywhere-convergent series Definition
• The residual r=Ax-μ x and the normwise backward error of an approximate eigenpair Definition
• The Tikhonov regularised least-squares objective ‖Ax-b‖₂²+λ‖x‖₂² for λ>0 Definition
• A diagonal stiff diffusion step with linear reaction Example
• A finite-matrix Schrödinger split preserves norm Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• Exact exponential stepping captures nonnormal transient growth Example
• Hilbert spaces are uniformly convex Example
• Norm-attaining functionals on a Hilbert space Example
• Positive coordinate weights define an inner product and change lengths and projections Example
• Real-time unitary and imaginary-time finite-matrix propagation Example
• The hexagonal A₂ cell: Euclidean cell metric versus graph distance Example
• A skew-Hermitian matrix exponential is unitary Lemma
• Certified action bounds control one exponential-integrator step Lemma
• Complex projector tangents are skew-Hermitian commutator directions Lemma
• Computed sensitivity errors perturb every singular value by at most their operator-norm error Lemma
• Exact logical-block correctness forces zero leakage Lemma
• Exponential Euler has a second-order local defect Lemma

…and 35 more results.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Sheldon Axler, Linear Algebra Done Right, 4th ed., §6A (standard reference, not scraped)
• Sergei Treil, Linear Algebra Done Wrong, Ch. 5, §5.1 (standard reference, not scraped)