The adjoint $T^{\ast }:W\to V$ is characterised by $⟨Tv,w{⟩}_W=⟨v,T^{\ast }w{⟩}_V$

Depends on

• Real and complex inner product spaces, with the inner product linear in the first argument
• Linear map between vector spaces over the same field

Used by

• Certified basis-pursuit denoising by primal-dual hybrid gradient Algorithm
• Chambolle–Pock primal-dual hybrid gradient Algorithm
• Distance corrects unlocated errors and known erasures Corollary
• Clean-ancilla zero-branch postselection Definition
• Clifford normalizer Definition
• Complementary channels and coherent information Definition
• Exact correction of a protected subsystem Definition
• Exact recovery of an encoded quantum code Definition
• Finite adiabatic path model Definition
• Finite block encoding Definition
• Finite Kraus representation Definition
• Finite projective measurements Definition
• Finite quantum mutual, conditional, and conditional mutual information Definition
• Finite-dimensional quantum erasure channel Definition
• Fixed-charge encoding and code projector Definition
• Galerkin coarse matrices and the exact coarse-correction error map Definition
• Graph adjacency and Laplacian generator conventions Definition
• Inverse Unitary Access Definition
• Isometric Stinespring representation Definition
• Matrix multiplicative-weights game and update Definition
• Normalized output law for finite quantum query branches Definition
• Operational quantum, private, and entanglement-assisted capacities Definition
• Projected-unitary encoding and singular vectors Definition
• Quantum acceptance probability for a finite promise computation Definition
• Quantum code as an isometric encoding and its code projector Definition
• Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space Definition
• Szegedy row-preparation map and discriminant Definition
• The Euclidean logarithmic norm of a complex matrix Definition
• The symmetric two-grid error map from a pre-smoother and Galerkin correction Definition
• Two-Level Unitary Definition
• Unitary conjugation map Definition
• A degenerate code with a non-diagonal Knill–Laflamme matrix Example
• A finite-matrix Schrödinger split preserves norm Example
• A projective measurement branch with probability zero Example
• Dirac and rank-one coordinate check Example
• Exact exponential stepping captures nonnormal transient growth Example
• Orthogonal error spaces and a recovery branch Example
• Product block error in a two-factor instance Example
• Quantum repetition subspace projector and a distance-one boundary error Example
• Qubit dephasing channel in Kraus and Choi forms Example

…and 95 more results.

Dependency tree · two levels

6 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., definition 7.1 (standard reference, not scraped)
• Sergei Treil, Linear Algebra Done Wrong, §5.5.1 (standard reference, not scraped)