• mathematical functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its...
    7 KB (1,275 words) - 00:14, 10 October 2023
  • Thumbnail for Isometry
    In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed...
    18 KB (2,425 words) - 20:13, 8 January 2025
  • an isometry when its action is restricted onto the support of A {\displaystyle A} , that is, it means that U {\displaystyle U} is a partial isometry. As...
    25 KB (4,218 words) - 12:36, 19 December 2024
  • complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator. The polar decomposition for matrices generalizes...
    12 KB (1,543 words) - 18:53, 21 August 2024
  • Thumbnail for Projection (linear algebra)
    {T}}} is the partial isometry that vanishes on the orthogonal complement of U {\displaystyle U} , and A {\displaystyle A} is the isometry that embeds U...
    34 KB (5,803 words) - 07:02, 14 December 2024
  • {\displaystyle U} such that U † U = I {\displaystyle U^{\dagger }U=I} (a partial isometry), the ensemble { q i , | φ i ⟩ } {\displaystyle \{q_{i},|\varphi _{i}\rangle...
    36 KB (5,310 words) - 11:52, 4 December 2024
  • Moore–Penrose pseudoinverse B+ can be. In that case, the operator B+A is a partial isometry, that is, a unitary operator from the range of T to itself. This can...
    29 KB (4,651 words) - 04:08, 1 January 2025
  • belonging to M are called (Murray–von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element...
    42 KB (5,912 words) - 03:52, 30 November 2024
  • and 1 − p are Murray–von Neumann equivalent, i.e., there exists a partial isometry u such that p = uu* and 1 − p = u*u. One can easily generalize this...
    13 KB (1,812 words) - 00:20, 24 September 2024
  • meaning that ee = e and e* = e. Every projection is a partial isometry, and for every partial isometry s, s*s and ss* are projections. If e and f are projections...
    25 KB (3,602 words) - 17:19, 7 January 2025
  • In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors...
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  • graphical representations of specific states, unitary operators, linear isometries, and projections in the computational basis | 0 ⟩ , | 1 ⟩ {\displaystyle...
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  • Thumbnail for Singular value decomposition
    bounded operator ⁠ M , {\displaystyle \mathbf {M} ,} ⁠ there exist a partial isometry ⁠ U , {\displaystyle \mathbf {U} ,} ⁠ a unitary ⁠ V , {\displaystyle...
    88 KB (14,054 words) - 17:40, 15 December 2024
  • Thumbnail for Riemannian manifold
    surface is called a local isometry. Call a property of a surface an intrinsic property if it is preserved by local isometries and call it an extrinsic...
    59 KB (8,681 words) - 00:11, 16 December 2024
  • operators is equivalent to finding unitary extensions of suitable partial isometries. Let H {\displaystyle H} be a Hilbert space. A linear operator A {\displaystyle...
    19 KB (3,248 words) - 14:42, 25 December 2024
  • the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous isometries of the manifold. More simply, the...
    27 KB (4,719 words) - 17:43, 29 December 2024
  • C*-algebras and k-graph C*-algebras are universal C*-algebras generated by partial isometries. The universal C*-algebra generated by a unitary element u has presentation...
    6 KB (976 words) - 10:27, 22 February 2021
  • kernel of P, clearly UP h = 0. But PU h = 0 as well. because U is a partial isometry whose initial space is closure of range P. Finally, the self-adjointness...
    4 KB (562 words) - 02:22, 1 March 2023
  • {\displaystyle \{x'_{k}:k<n\}} ). The union of these maps defines a partial isometry ϕ : X → X ′ {\displaystyle \phi :X\to X'} whose domain resp. range...
    3 KB (402 words) - 18:43, 27 November 2024
  • Thumbnail for Itô calculus
    Itô isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itô isometry. The...
    30 KB (4,486 words) - 16:45, 26 November 2024
  • {\displaystyle T(x_{1},x_{2},x_{3},\dots )=(x_{2},x_{3},x_{4},\dots ).} T is a partial isometry with operator norm 1. So σ(T) lies in the closed unit disk of the complex...
    26 KB (3,810 words) - 21:19, 1 October 2024
  • V2, K2) be two Stinespring representations of a given Φ. Define a partial isometry W : K1 → K2 by W π 1 ( a ) V 1 h = π 2 ( a ) V 2 h . {\displaystyle...
    12 KB (2,113 words) - 06:14, 30 June 2023
  • structure of T {\displaystyle T} means that a "truncated" shift is a partial isometry on H {\displaystyle {\mathcal {H}}} . More specifically, let { e 0...
    7 KB (1,111 words) - 09:50, 8 January 2025
  • group of isometries of X {\displaystyle X} acts by homeomorphisms on ∂ X {\displaystyle \partial X} . This action can be used to classify isometries according...
    20 KB (3,139 words) - 15:28, 4 December 2024
  • Thumbnail for Terence Tao
    introduced the notion of a "restricted linear isometry," which is a matrix that is quantitatively close to an isometry when restricted to certain subspaces.[CT05]...
    79 KB (6,683 words) - 20:58, 1 January 2025
  • decomposition A = V | A | , {\displaystyle A=V|A|,\,} it says that the partial isometry V should lie in M and that the positive self-adjoint operator |A| should...
    6 KB (800 words) - 15:36, 3 November 2019
  • e : e ∈ E 1 } {\displaystyle \left\{s_{e}:e\in E^{1}\right\}} are partial isometries with mutually orthogonal ranges, the elements of { p v : v ∈ E 0 }...
    26 KB (4,543 words) - 11:37, 2 January 2025
  • Thumbnail for Gaussian curvature
    surface S in R3. A local isometry is a diffeomorphism f : U → V between open regions of R3 whose restriction to S ∩ U is an isometry onto its image. Theorema...
    19 KB (2,632 words) - 08:54, 18 November 2024
  • Thumbnail for Symmetry (physics)
    spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed...
    27 KB (3,283 words) - 23:35, 3 January 2025
  • Hines, Peter; Braunstein, Samuel L. (2010). "The Structure of Partial Isometries". In Gay and, Simon; Mackie, Ian (eds.). Semantic Techniques in Quantum...
    28 KB (3,748 words) - 02:47, 3 May 2024