Dual finite frames for vector spaces over an arbitrary field with applications

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

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Linear Algebra and the Dirac Notation

An Introduction to Quantum Computing ◽

10.1093/oso/9780198570004.003.0005 ◽

2006 ◽

Author(s):

Phillip Kaye ◽

Raymond Laflamme ◽

Michele Mosca

Keyword(s):

Hilbert Space ◽

Quantum Mechanics ◽

Quantum Computing ◽

Linear Algebra ◽

Hilbert Spaces ◽

Vector Spaces ◽

Complex Vector ◽

Dual Vector ◽

Finite Dimensional ◽

Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

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Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1978-004-0 ◽

1978 ◽

Vol 21 (1) ◽

pp. 17-19

Author(s):

Dragomir Ž. Djoković

Keyword(s):

Tensor Product ◽

Hilbert Spaces ◽

Unitary Representations ◽

Vector Spaces ◽

Closed Field ◽

Discrete Groups ◽

Algebraically Closed Field ◽

Finite Dimensional ◽

Irreducible Unitary Representations

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

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Frame for operators in finite dimensional hilbert space

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.49.2018.2383 ◽

2018 ◽

Vol 49 (1) ◽

pp. 35-48

Author(s):

Mohammad Janfada ◽

Vahid Reza Morshedi ◽

Rajabali Kamyabi Gol

Keyword(s):

Hilbert Space ◽

Hilbert Spaces ◽

Dual Frames ◽

Finite Dimensional ◽

Dimensional Hilbert Space ◽

Finite Dimensional Hilbert Space

In this paper, we study frames for operators ($K$-frames) in finite dimensional Hilbert spaces and express the dual of $K$-frames. Some properties of $K$-dual frames are investigated. Furthermore, the notion of their oblique $K$-dual and some properties are presented.

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Multi-variable quaternionic spectral analysis

Opuscula Mathematica ◽

10.7494/opmath.2021.41.3.335 ◽

2021 ◽

Vol 41 (3) ◽

pp. 335-379

Author(s):

Ilwoo Cho ◽

Palle E.T. Jorgensen

Keyword(s):

Spectral Analysis ◽

Functional Equations ◽

Linear Functional ◽

Vector Spaces ◽

Complex Numbers ◽

Dimensional Vector ◽

Finite Dimensional ◽

Linear Functional Equations ◽

Non Linear

In this paper, we consider finite dimensional vector spaces $\mathbb{H}^n$ over the ring $\mathbb{H}$ of all quaternions. In particular, we are interested in certain functions acting on $\mathbb{H}^n$, and corresponding functional equations. Our main results show that (i) all quaternions of $\mathbb{H}$ are classified by the spectra of their realizations under representation, (ii) all vectors of $\mathbb{H}^n$ are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring $M_n(\mathbb{C})$ of all $(n \times n)$-matrices over the complex numbers $\mathbb{C}$ has close connections with certain "non-linear" functional equations on $\mathbb{H}^n$ up to the classification of (ii).

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Quantum Mechanics as Hamilton–Killing Flows on a Statistical Manifold

Physical Sciences Forum ◽

10.3390/psf2021003012 ◽

2021 ◽

Vol 3 (1) ◽

pp. 12

Author(s):

Ariel Caticha

Keyword(s):

Quantum Mechanics ◽

Hilbert Spaces ◽

Cotangent Bundle ◽

Symplectic Structure ◽

Information Geometry ◽

Complex Numbers ◽

Born Rule ◽

Statistical Manifold ◽

Finite Dimensional ◽

Starting Point

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.

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Plane polygons revisited

Journal of Applied Probability ◽

10.2307/3213554 ◽

1982 ◽

Vol 19 (A) ◽

pp. 113-122 ◽

Cited By ~ 1

Author(s):

B. H. Neumann

Keyword(s):

Vector Spaces ◽

Complex Numbers ◽

Dimensional Vector ◽

Finite Dimensional ◽

Electrical Engineers ◽

Equilateral Triangles ◽

Arbitrary Plane ◽

Arbitrary Triangle ◽

Napoleon’S Theorem ◽

Current Systems

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

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OBLIQUE DUAL FRAMES IN FINITE-DIMENSIONAL HILBERT SPACES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691313500112 ◽

2013 ◽

Vol 11 (02) ◽

pp. 1350011 ◽

Cited By ~ 2

Author(s):

XIANG-CHUN XIAO ◽

YU-CAN ZHU ◽

XIAO-MING ZENG

Keyword(s):

Hilbert Spaces ◽

Computational Efficiency ◽

Tight Frame ◽

Constructive Method ◽

Dual Frame ◽

Dual Frames ◽

Finite Dimensional ◽

Oblique Dual

A frame can be completed to a tight frame by adding some additional vectors. However, for the purpose of computational efficiency, we need to put restrictions to the number of added vectors. In this paper we propose a constructive method that allows to extend a given frame to an oblique dual frame pair such that the number of added vectors is in general much smaller than the number of added vectors used to extend it to a tight frame. We also present a uniqueness characterization and several equivalent characterizations for an oblique dual frame pair, it turns out that oblique dual frame pair provides more flexibilities than alternate dual frame pair.

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Chapter 6: Finite-Dimensional Hilbert Spaces and the Matrix Formulation of the Conjugate Gradient Method

Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs ◽

10.1137/1.9781611973846.ch6 ◽

2014 ◽

pp. 47-51

Keyword(s):

Conjugate Gradient Method ◽

Hilbert Spaces ◽

Conjugate Gradient ◽

Gradient Method ◽

Matrix Formulation ◽

Finite Dimensional ◽

The Matrix

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Plane polygons revisited

Journal of Applied Probability ◽

10.1017/s0021900200034501 ◽

1982 ◽

Vol 19 (A) ◽

pp. 113-122 ◽

Cited By ~ 2

Author(s):

B. H. Neumann

Keyword(s):

Vector Spaces ◽

Complex Numbers ◽

Dimensional Vector ◽

Finite Dimensional ◽

Electrical Engineers ◽

Equilateral Triangles ◽

Arbitrary Plane ◽

Arbitrary Triangle ◽

Napoleon’S Theorem ◽

Current Systems

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

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