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Identities of Vector Algebra as Associative Properties of Multiplicative Compositions of Quaternion Matrices11 Victor Kravets1,a, Tamila Kravets1, Olexiy Burov2 1
National Mining University, Dnipro, Ukraine
2
Jack Baskin School of Engineering, University of California-Santa Cruz, CA, USA
a
prof.w.kravets@gmail.com DOI 10.2412/mmse.47.87.900 provided by Seo4U.link
Keywords: quaternionic matrices, vector matrices, vector algebra identities, complex vector and scalar products, associative property of vector matrices products.
ABSTRACT. This paper is dedicated to the further development of matrix calculation in the sphere of quaternionic matrices. Mathematical description of transfer (displacement) and turn (rotation) in space are fundamental for the mechanics of rigid body. The transfer (displacement) in space is described by the vector (hodograph). The turn (rotation) in space is described by quaternion. Calculation quaternionic matrices generalizes vector algebra and is directly adapted for the computing experiment concerning nonlinear dynamics of discrete mechanical systems in spatial motion. It is proposed to examine the turn and transfer of the rigid body in space with four-dimensional orthonormal basis and corresponding matrices equivalent to quaternions or vectors. The identities of vector algebra, including the Lagrange identity, Euler-Lagrange identity, Gram determinant and others, are found systematically. The associative products of conjugate quaternionic matrices are represented by the multiplicative compositions of vector algebra. The complex vector and scalar products are represented by the introduced range of vector algebra identical equations is found, including the known ones, which serve, in particular, to justify the fidelity of the offered method. The method is being developed to represent associative products of conjugate and various quaternionic matrices by multiplicative compositions of vector algebra, containing scalar and vector products. The method is offered to represent complex (vector and scalar) vector algebra products as quaternionic matrices. This fundamental results constitute the first part of the study recommended for engineers, high school teachers and students who in their practical activity set and solve the problems of dynamic design of aeronautical engineering, rocket engineering, space engineering, land transport (railway and highway transport), robotics, etc. and exposed data to be able to contribute to the research area, to permit to enhance the intellectual performance, to provide the engineer with simple and efficient mathematical apparatus.
Introduction. In computational experiment for nonlinear dynamics of discrete mechanical systems satisfied by an appropriate choice of variables and new organization of calculation process. The organization of calculation process is defined by the mathematical model of technical specification formed on the ground of traditional mathematical apparatus: vector algebra, quaternion algebra, calculus of tensors, theory of screws, matrix calculus, and, in particular, matrices equivalent to quaternions. Particular quaternionic matrices R. Bellman [1], A. Maltsev [2], G. Korenev [3], N. Kilchevskiy [4]. Quaternionic matrices found application in exposing the principles of symmetry in physics [5, 6], in numerical calculation for attitude control [7, 8], in relative quaternions calculation [9], in theory of finite rotation [10, 11], in theory of inertial guidance [12, 13], in kinematics and dynamics of solid bodies [14 - 18]. Yet, methodical fundamental research for quaternionic matrices 11
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representation for problem solution in analytical dynamics is basic, and computer technologies application implies the introduction of an established reference system and the reduction of esearch is motivated and justified, being focused on efficiency improvement of computational experiment in problems The work objective is to work out the quaternionic matrices nd their application in vector algebra as a tool for designing symbolic models exclusively adapted for modern computer technologies of calculation experiment [19, 20]. xperiment the finite set of quartic monomial matrices is established, which is put in correspondence with the four-dimensional orthonormal basis and creates a multiplicative group. Two noncommutative octic subgroups are fixed on set of monomial matrices isomorphous to the group of quaternions [21]. It is demonstrated that the introduced set of four quatric quaternionic matrices is isomorphous to the quaternions algebra and generalizes the vector algebra on the plane and in three-dimensional space. The formulas are found for matrix representation of the set of complex vector and scalar products of vector algebra. The suggested method of matrix representation of complex vector and scalar products permits also to find systematically the vector algebra identities correlated with the known results, including the Lagrange identity, Euler-Lagrange identity, Gram determinant and other formulas. Formulation of the problem Applying mathematical induction, the procedure of vector representation of associative products for various and conjugate quaternionic matrices is established. Expanded symbolic formulas are found, which vector algebra multiplicative compositions. regarded as an inverse one. Detailed procedures are provided, and the expanded symbolic formulas are established. The formulas reflect the equivalent correspondences of the set of the quaternionic aining complex vector and scalar products of several vectors. of conjugate quaternionic matrices are examined. For the examined compositions, vector repres combinations. The vector algebra identical equations are set, which are defined by the matrix
The quaternionic matrices defined by the basis with a scalar part equal to zero are examined, i.e., equivalent to vector and opposite to vector:
,
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The vector in three-dimensional space is represented as a matrix
(column vector):
or
for two vectors:
1. 2.
for three vectors:
for four vectors we obtain respectively:
2. A0
A0t B0
B0t C0 C0t d 0
A0 B0C0d 0
A0 B0C0t d 0
A0 B0t C0d 0
A0 B0t C0t d 0
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A0t B0C0d 0
A0t B0C0t d 0
A0t B0t C0d 0
A0t B0t C0t d 0 ,
Mechanics, Materials Science & Engineering, March 2017
and so on for five and more vectors.
multiplicative combination is the only one:
1.1. 1.2.
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possible:
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1.8.
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2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
3.1.
3.2.
3.3.
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3.4.
3.5.
3.6.
3.7.
3.8.
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
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4.7.
4.8.
5.1.
5.2.
5.3.
5.4.
5.5
5.6.
5.7.
5.8
and so on for five and more vectors.
combinations
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appropriate multiplicative vector algebra compositions containing scalar and vector products are found. For two vectors, we obtain:
1.1.
1.2.
For three vectors, we obtain respectively:
and
It is convenient to systematize as a table the results provided here (Table 1).
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Table 1. Calculation results.
= 1.
+
-
-
+
2.
+
-
+
-
3.
+
+
-
-
4.
+
+
+
+
+
-
-
+
+
-
+
-
+
+
-
-
+
+
+
+
+
-
-
+
+
-
+
-
+
+
-
-
+
+
+
+
1.1.
1.2.
1.3.
1.4.
2.1.
2.2.
2.3.
4
2.4.
a b
0 c b a c
For four vectors the results obtained are systematized in form of tables (Table 2 Table 7). Table 2. Multiplication results. MMSE Journal. Open Access www.mmse.xyz
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1
+
+
+
+
+
+
2
+
+
-
-
+
+
3
+
-
+
-
+
-
4
+
+
+
+
-
-
5
+
-
-
+
+
-
6
+
+
-
-
-
-
7
+
-
+
-
-
+
8
+
-
-
+
-
+
= 1
+
+
2
-
-
3
+
-
4
-
-
5
-
+
6
+
+
7
-
+
8
+
-
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Table 3. Multiplication results.
1.1
+
+
+
+
+
+
1.2
+
+
-
-
+
+
1.3
+
-
+
-
+
-
1.4
+
+
+
+
-
-
1.5
+
-
-
+
+
-
1.6
+
+
-
-
-
-
1.7
+
-
+
-
-
+
1.8
+
-
-
+
-
+
1.1
+
+
1.2
-
-
1.3
+
-
1.4
-
-
1.5
-
+
1.6
+
+
a b 1.7
-
+
1.8
+
-
8
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c d 0
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Table 4. Multiplication results.
2.1
+
+
+
+
+
+
2.2
+
+
-
-
+
+
2.3
+
-
+
-
+
-
2.4
+
+
+
+
-
-
2.5
+
-
-
+
+
-
2.6
+
+
-
-
-
-
2.7
+
-
+
-
-
+
2.8
+
-
-
+
-
+
2.1
+
+
2.2
-
-
2.3
+
-
2.4
-
-
2.5
-
+
2.6
+
+
2.7
-
+
2.8
+
-
8
0 a c b d
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a d b c
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Table 5. Multiplication results.
3.1
+
+
+
+
+
+
3.2
+
+
-
-
+
+
3.3
+
-
+
-
+
-
3.4
+
+
+
+
-
-
3.5
+
-
-
+
+
-
3.6
+
+
-
-
-
-
3.7
+
-
+
-
-
+
3.8
+
-
-
+
-
+
3.1
+
+
3.2
-
-
3.3
+
-
3.4
-
-
0
8 a b d c
3.5
-
+
3.6
+
+
3.7
-
+
3.8
+
-
a c d b
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a b cd
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Table 6. Multiplication results.
4.1
+
+
+
+
+
+
4.2
+
+
-
-
+
+
4.3
+
-
+
-
+
-
4.4
+
+
+
+
-
-
4.5
+
-
-
+
+
-
4.6
+
+
-
-
-
-
4.7
+
-
+
-
-
+
4.8
+
-
-
+
-
+
4.1
+
+
4.2
-
-
4.3
+
-
4.4
-
-
4.5
-
+
4.6
+
+
4.7
-
+
4.8.
+
-
Table 7. Multiplication results. MMSE Journal. Open Access www.mmse.xyz
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5.1
+
+
+
+
+
+
5.2
+
+
-
-
+
+
5.3
+
-
+
-
+
-
5.4
+
+
+
+
-
-
5.5
+
-
-
+
+
-
5.6
+
+
-
-
-
-
5.7
+
-
+
-
-
+
5.8
+
-
-
+
-
+
5.1
+
+
5.2
-
-
5.3
+
-
5.4
-
-
5.5
-
+
5.6
+
+
5.7
-
+
5.8
+
-
And so on for five and more vectors.
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multiplicative compositions we find the following vector and matrix correspondences for two vectors:
,
For three vectors:
and
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Or
For four vectors:
1.
2.
3.
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4.
5. 6.
7.
8.
And so on for five and more vectors. Setting vector algebra identical equations algebra equalities for three vectors are set: MMSE Journal. Open Access www.mmse.xyz
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1.
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; 2.
; 3.
;
4.
.
Whence it follows:
or, using a second-order determinant, we obtain formulas in ordered recording:
For four vectors from the first group of equalities we obtain:
1.1.1. 1.1.2. 1.1.3.
i.e.,
1.1.1. a
b
c
d
a b
c d
or
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a c
b d
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Whence it follows:
1.1.3.
From the second group of equalities we obtain:
2.1.1. 2.1.2.
2.1.3.
2.2.1.
2.2.2.
And
2.3.1.
Whence it follows:
2.1.1.
Or
2.1.2.
a b
c
d
[ a b
d ]c
a b (c d )
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Or
2.1.3.
Or
and also
2.2.1.
Or
2.2.2.
Or
2.3.1.
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Or
The third group of equalities is a trivial one. From the fourth group of equalities we obtain:
4.1.1. 4.1.2. 4.1.3.
and also
4.2.1. 4.2.2.
And
4.3.1.
Whence it follows:
4.1.2.
Or
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and also
4.1.3.
or
4.3.1.
Or
The fifth and sixth groups of equalities are trivial. From the seventh group of equalities we obtain:
7.1.1. 7.1.2. 7.1.3.
and also
7.2.1.
a b
c
d
a c b d
b c
7.2.2.
And
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a d ;
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7.3.1.
Whence it follows:
7.1.1.
7.2.1.
7.3.1.
From the eighth group of equities we obtain:
8.1.1. 8.1.2. 8.1.3.
and also
8.2.1. 8.2.2. 8.3.1.
Whence it follows:
Summarizing, we point out that some of the provided results correspond to the Lagrange identity:
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particularly, to
ISSN 2412-5954
, the Euler-Lagrange formula:
The Gramian matrix:
and other known correlations [22]. Construct of the equalities connecting five and more vectors is obvious but is lengthy for representation. Summary. The research objective is to elaborate the elements of calculation of quaternionic matrices and their application to vector algebra. The problems set: to find the equivalent correspondences to associative products of quaternionic matrices and vector algebra multiplicative compositions; to represent complex vector and scalar vector algebra products with quaternionic matrices; to represent complex vector and scalar vector algebra products with identical equations are solved. algorithms as quaternionic matrices acquire symmetry, invariant property, compactness and universality, which makes the programming faster, facilitates verification and makes scientific research more convenient, i.e. improves the productivity of intellectual labor. References [1] Bellman, R. Introduction to matrix analysis, Second edition, Book code: CL19, Series: Classics in Applied Mathematics, 1997, 403 p. DOI: http://dx.doi.org/10.1137/1.9781611971170. [2] Mal'tsev, A.I. Osnovy linejnoj algebry [Fundamentals of linear algebra], Moscow, Nauka Publ., 1970, 400 p. (in Russian). [3] Korenev, G.V. Tenzornoe ischislenie [Tensor calculus], MFTI Publ., 1995, 240 p. (in Russian). [4] Kil'chevskij, N.A. Kurs teoreticheskoj mexaniki [Course of theoretical mechanics], Moscow, Nauka Publ., 1977, Part 1, 480 p., Part 2, 544 p. (in Russian). [5] Elliot, J.P., Dawber, P.G. Symmetry in physics, Vol. 1: Principles and Simple Applications, Oxford University Press, 1985, 302 p., Vol. 2: Further Applications, Oxford University Press, 1985, 298 p. [6] Berezin, A.V., Kurochkin, Yu.A., Tolkachev, E.A. Kvaterniony v relyativistskoj fizike [Quaternions in relativistic physics], Moscow, Editoreal Publ., 2003, 200 p. (in Russian). [7] Branets, V.N., Shmyglevskij, I.P. Primenenie kvaternionov v zadachax orientatsii tverdogo tela [The use of quaternions in problems of solid-state orientation], Moscow, Nauka Publ., 1973, 320 p. (in Russian).
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[8] Raushenbax, B.V., Tokar', E.N. Upravlenie orientatsiej kosmicheskix apparatov [The orientation of the spacecraft management], Moscow, Nauka Publ., 1974, 600 p. (in Russian). [9] Mejo, R.A. Perexodnaya matritsa dlya vychisleniya otnositel'nyx kvaternionov [The transition matrix to calculate the relative quaternion], Raketnaya texnika i kosmonavtika [Rocketry and Astronautics], 17 (3), 1979, p. 184-189. (in Russian). [10] Lur'e, A.I. Analiticheskaya mexanika [Analytical mechanics], Moscow, Fizmatgiz Publ., 1961, 824 p. (in Russian). [11] Plotnikov, P. K., Chelnokov, Yu. N. Primenenie kvaternionnyx matrits v teorii konechnogo povorota tverdogo tela [Application of quaternion matrices in the final turn in solid state theory], Sbornik nauchn.-metod. state [12] Ishlinskij, A.Yu. Orientatsiya, giroskopy i inertsial'naya navigatsiya [Orientation, gyroscopes and inertial navigation], Moscow, Nauka Publ., 1976, 672 p. (in Russian). [13] Onischenko, S.M. Primenenie giperkompleksnyx chisel v teorii inertsial'noj navigatsii. Avtonomnye sistemy [The use of hyper complex numbers in the inertial navigation theory. Standalone systems], Kyiv, Naukova dumka Publ., 1983, 208 p. (in Russian). [14] Pars, L.A. A Treatise on analytical dynamics, Ox Bow Pr. Publ., 1981, 641 p. [15] Chelnokov, Yu.N. Kvaternionnye i bikvaternionnye modeli i metody mexaniki tverdogo tela i ix prilozheniya. Geometriya i kinematika dvizheniya [Quaternion and biquaternions models and methods of solid mechanics and their applications. The geometry and kinematics motion], Moscow, Fizmatgiz Publ., 2006, 512 p. (in Russian). [16] Kravets, T.V. Based on Gibbs vector solution of spatial angular stabilization of solid problem, Avtomatika 2001: Sbornik nauchnyx trudov konferentsii [Automation 2001: Proceedings of the Conference], Odessa Publ., 2001, Volume 2, p. 20. [17] Kravets, V.V., Kravets, T.V. On the nonlinear dynamics of elastically interacting asymmetric rigid bodies, Int. Appl. Mech., 2006, 42(1), p. 110- 114. [18] Kravets, V.V., Kravets, T.V., Kharchenko, A.V. Using quaternion matrices to describe the kinematics and nonlinear dynamics of an asymmetric rigid body, Int. Appl. Mech., 2009, 45 (223), DOI: 10.1007/s10778-009-0171-1. [19] Pivnyak, G.G., Kravets, V.V., Bas, K.M., Kravets, T.V., Tokar', L.A. Elements of calculus quaternionic matrices and some applications in vector algebra and kinematics, MMSE Journal, 3, March 2016, p.p. 46-56. ISSN 2412-5954, Open access www.mmse.xyz, DOI 10.13140/RG.2.1.1165.0329. [20] Kravets, V.V., Bass, K.M., Kravets, T.V., Tokar L.A. Dynamic design of ground transport with the help of computational experiment, MMSE Journal, 1, October 2015, p.p. 105 - 111. ISSN 24125954, Open access www.mmse.xyz, DOI 10.13140/RG.2.1.2466.6643. [21] Kravets, V., Kravets, T., Burov, O. Monomial (1, 0, -1)-matricesthe transfer in space. Lap Lambert Academic Publishing, Omni Scriptum GmbH&Co. KG., 2016, 137 p. ISBN: 978-3-330-01784-9 [22] Korn, G., Korn, T. Spravochnik po matematike dlya nauchnyh rabotnikov i inzhenerov [Mathematical Handbook for Scientists and Engineers], Moscow, Nauka Publ., 1984, 832 p. (in Russian). Cite the paper Victor Kravets, Tamila Kravets, Olexiy Burov (2017). Identities of Vector Algebra as Associative Properties of Multiplicative Compositions of Quaternion Matrices. Mechanics, Materials Science & Engineering, Vol 8. doi:10.2412/mmse.47.87.900
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