Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces

Depends on

• Real and complex inner product spaces, with the inner product linear in the first argument
• The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product
• Linear map between vector spaces over the same field

Used by

• Isometry Unitary Completion Algorithm
• Diameter rigidity from toponogov under a sectional lower bound Corollary
• Lebesgue measure on ℝⁿ is invariant under every orthogonal linear map Corollary
• A Hecke parameter does not certify a positive unitary structure Counterexample
• A preserved unknown state cannot program its own reflection Counterexample
• Algebraic braid relations do not imply unitarity Counterexample
• Clean-ancilla zero-branch postselection Definition
• Clifford normalizer Definition
• Complex and real generalized Schur forms Definition
• Controlled Unitary Access Definition
• Exact one-qubit gates and rotations Definition
• Finite block encoding Definition
• Fixed-charge encoding and code projector Definition
• Hilbert direct sums of unitary representations Definition
• Isometric Stinespring representation Definition
• Left and right regular unitary representations of an LCH group Definition
• Multi-Control Activation Pattern Definition
• Normalized output law for finite quantum query branches 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
• Real and complex Schur forms Definition
• Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator Definition
• Szegedy row-preparation map and discriminant Definition
• Two-Level Unitary Definition
• Unitary conjugation map Definition
• A circle representation with an averaged orthogonal weight form Example
• A moved-space intersection in A₃ that is not the meet Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• Isotypic Haar projections specialize to finite character sums Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Newtonian potential of radial compact data Example
• Product block error in a two-factor instance Example
• Simple and reflection lengths of a long transposition in S₅ Example
• The Wall form and line restrictions of a plane rotation, and the necessity of a common upper bound Example
• Two-term LCU normalization and access count Example
• Worked Fibonacci and Ising braid matrices Example
• A finite isometry extends after finite ancilla padding Lemma
• A Hurwitz matrix exponential has a polynomial-exponential decay bound Lemma
• A local gate has a unique global wire embedding Lemma

…and 50 more results.

Dependency tree · two levels

8 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., definitions 7.44 and 7.51 (standard reference, not scraped)