Matrix representation of quaternions

Farebrother, R. W.; Groß, Jürgen; Troschke, Sven-Oliver

doi:10.1016/s0024-3795(02)00535-9

Cited by 38 publications

(26 citation statements)

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“…The multiplication of two rotation matrices can also be expressed a as a multiplication of their corresponding quaternions according to the multiplication rule [55]…”

Section: Discussionmentioning

confidence: 99%

Hamiltonians and order parameters for crystals of orientable molecules

2018

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Many modern functional materials exhibit microscopic degrees of freedom that are difficult to describe mathematically or do not conform to conventional models in solid state physics. In this paper, we consider crystals containing orientable molecular species, which encompass the class of promising photovoltaic materials CH 3 NH 3 PbX 3 . Abstracting these materials as crystals of orientable rigid rotors, we develop an effective Hamiltonian that expresses the crystal potential energy in terms of collective orientations of interacting rigid rotors. The approach is motivated by the cluster expansion framework from alloy theory and is appropriate for describing both chiral and achiral molecules. This framework utilizes a quaternion parameterization of molecular orientation and makes full use of the symmetry both of the rotors and of the crystal in order to constrain the functional form of the final Hamiltonian expression. The resulting Hamiltonian is compact, systematically improvable, and is suitable for Monte Carlo or molecular dynamics simulation.We apply this formalism to CH 3 NH 3 PbI 3 and report symmetry-adapted basis functions that can be used to construct order parameters or Hamiltonians that describe orientational correlation of methylammonium ions in hybrid organic-inorganic halide perovskite materials.

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“…The multiplication of two rotation matrices can also be expressed a as a multiplication of their corresponding quaternions according to the multiplication rule [55]…”

Section: Discussionmentioning

confidence: 99%

Hamiltonians and order parameters for crystals of orientable molecules

2018

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“…It was proven in [4] that there are 48 distinct representations of quaternions by skew-symmetric signed permutation matrices. More precisely, H, J, K can be chosen from two sets {±X s , ±Y s , ±Z s }, s = 1, 2 under assumption that we take either a matrix or its negation and formulae (2) are satisfied where…”

Section: Adv Appl Clifford Algebrasmentioning

confidence: 99%

On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis

Bajorska-Harapińska

Smoleń

Wituła

2019

Adv. Appl. Clifford Algebras

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This paper is devoted to the newly defined families of associated sequences of real polynomials and numbers that arose on a base of quaternions. We present not only the explicit and recurrent formulae for these sequences, but also summation and reduction ones. Some (but not all) of these sequences can be found in OEIS. Moreover, the obtained results also speak a lot about the quaternions structure itself. Also, while examining matrices related to these sequences we discovered some general formulae for powers of so-called arrowhead matrices.

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“…In this case col(a) must be defined as the first column of ı 1 (a). In a paper by Farebrother, Groß, and Troschke, [2], these authors in 2003 have made a systematic search for all matrix representations of H in R 4×4 . All rules (2.13) to (2.15) are immediate consequences of (2.11), (2.12).…”

Section: Quaternions and Pseudoquaternions In The Matrix Space R 4×4mentioning

confidence: 99%

The Nonexistence of Pseudoquaternions in $${\mathbb{C}^{2\times 2}}$$

Janovská

Opfer

2010

Adv. Appl. Clifford Algebras

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The field of quaternions, denoted by H can be represented as an isomorphic four dimensional subspace of R 4×4 , the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in R 4×4 which is also a field and which has -in connection with the quaternions -many pleasant properties. This field is called field of pseudoquaternions. It exists in R 4×4 but not in H. It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in R 4 . And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b. Now, the field of quaternions can also be represented as an isomorphic four dimensional subspace of C 2×2 over R, the space of complex matrices with two rows and columns. We show that in this space pseudoquaternions with all the properties known from R 4×4 do not exist. However, there is a subset of C 2×2 for which some of the properties are still valid. By means of the Kronecker product we show that there is a matrix in C 4×4 which has the properties of the pseudoquaternionic matrix.Mathematics Subject Classification (2010). 11R52, 12E15.

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Matrix representation of quaternions

Cited by 38 publications

References 1 publication

Hamiltonians and order parameters for crystals of orientable molecules

Hamiltonians and order parameters for crystals of orientable molecules

On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis

The Nonexistence of Pseudoquaternions in $${\mathbb{C}^{2\times 2}}$$

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