Tensor Trigonometry. – Moscow: Fizmatlit, 2021, 320 p.

11.3 MB·English

by  Ninul A. S.

Ninu´l A. S.
TENSOR TRIGONOMETRY
Publisher “FIZMATLIT”
ББК 22.143
Н 60
UDC 514.1/512.64/530.12
MSC2020: 15A/51F/53B/83(ofAMSandzbMATH)
Ninu´l A. S. TensorTrigonometry. −Moscow: ScientificPublisher“FIZMATLIT”,2021,320p.,8ill.
(Englishauthorversionofthe1-stRussianedition: Ninu´lA.S.TensorTrigonometry. TheoryandApplication.
−Moscow: “MIR” Publisher,2004,336p. withtheadditionof"Kunstkammer".)
Planimetryincludesmetricpartandtrigonometry. IngeometriesofmetricspacesfromtheendofXIXage
theirtensorformsarewidelyused. Howeverthetrigonometryisremainedonlyinitsscalarforminaplane.
The tensor trigonometry is development of the flat scalar trigonometry from Leonard Euler classic forms
into general multi-dimensional tensor forms with vector and scalar orthoprojections and with step by step
increasingacomplexityandopportunities. Describedinthebookarefundamentalsofthisnewmathematical
subjectwithmanyinitialexamplesofitsapplications.
Intheoreticplan,thetensortrigonometrycomplementsnaturallyAnalyticGeometryandLinearAlgebra.
In practical plan, it gives the clear instrument for solutions of various geometric and physical problems in
homogeneousisotropicspaces,suchasEuclidean,quasi-andpseudo-Euclideanones. Sointhesespaces,the
tensortrigonometrygivesverycleargenerallawsofmotionsincompleteformsandwithpolardecompositions
into principal and secondary motions, their descriptive trigonometric vector models, which are applicable
also to n-dimensional non-Euclidean geometries in subspaces of constant radius embedded into enveloping
metricspaces,andinthetheoryofrelativity. InSTR,theapplicationswereconsideredtillatrigonometric
pseudo-analog of the classic theory by Frenet–Serret with absolute differentially-geometric, kinematic and
dynamiccharacteristicsinthecurrentpointsofaworldline.
Newmethodsofthetensortrigonometrycanbealsousefulinotherdomainsofmathematicsandphysics.
Thebookisintendedforresearchersinthefieldsofmulti-dimensionalspaces,analyticgeometry,linearalgebra
withtheoryofmatrices,non-Euclideangeometries,theoryofrelativityandtoallthosewhoisinterestedin
new knowledges and applications, given by exact sciences. It may be useful for educational purposes with
thisnewmathsubjectintheuniversitydepartmentsofalgebra,geometryandphysics.
ISBN-13 : 978-5-94052-278-2(Izdatel‘stvofiziko-matematicheskojliteratury‘)
DOI 10.32986/978-5-94052-278-2-320-01-2021
The first edition of the book (ISBN-10 : 5-03-003717-9 and ISBN-13 : 978-5-03-003717-2) was reviewed
by professor of Moscow State University, Dr.Sc. M. M. Postnikov and by professor of Moscow Regional
University,Dr.Sc. O.V.Manturov.
All rights reserved.
Copyright (cid:13)c 2021, 2004 by Anatoly S. Ninul
ThefirsteditioninRussianofthebookinSeptember2004wasdevotedbytheauthor
tothe175ysanniversaryofthefirstpublicationsonnon-EuclideanGeometry,
tothe100ysanniversaryofthefirstpublicationsonTheoryofRelativity
andtotheirgreatcreators–Lobachevsky,Bolyai,Lorentz,Poincar´e,Einstein
To the readers
Seldom, what division of mathematical science is so well-known and understandable yet
since school years as the Trigonometry. Originated in antiquity it practically completed own
developmentandobtaineditsmodernformattheendofthe18thcenturyintheworksofgreat
LeonardEuler. MeanwhileGeometry, fromthehistoricallyinitialEuclideanforms, passedfar
aheadforthelasttwocenturies. Furthermore,itsvariousmulti-dimensionalandnon-Euclidean
tensor forms were discovered and studied.
In the monograph, we undertake constructing general forms of the TensorTrigonometry
in multi-dimensional homogeneous and isotropic spaces with quadratic metrics (as Euclidean,
quasi-andpseudo-Euclideanones). TheclassicScalarTrigonometryactsoneigenplanesofthe
binary trigonometric subspace of a tensor angle. The angle between two lines (or vectors),
between two subspaces (or lineors) in multi-dimensional linear spaces has accordingly the
natureofbivalenttensors,determinedbythesetreflectortensorofthebinaryspace. However,
itskindisdeterminedbytheconcretequadraticmetric. Inthesemetricspaces,atensorangle
and its trigonometric functions are respectively either orthogonal, or quasi-orthogonal, or
pseudo-orthogonal tensors. (In particular, for Euclidean spaces, the simplest reflector tensor
isaunitymatrix,andwecandealonlywiththemiddlereflectoroftheconcretetensorangle.)
These tensor angles and all their trigonometric functions can be defined in the two forms:
(1) projective one by a pair of eigenprojectors or eigenreflectors; (2) motive one by the given
rotational or deformational matrix. Projective and motive angles are one-to-one connected.
In order to obtain the tensor construction, it was necessary to consider highly thoroughly
anumberofrelatedquestionsintheTheoryofExactMatrices,whatisapartofLinearAlgebra.
In addition to this, our efforts were rewarded by attainments of interesting and unexpected
results in Algebra, Geometry and Theoretical Physics.
TensorTrigonometrypointofviewgivessuchadvantages,thatsomeratherdifficultandnot
easily perceivable mathematical or physical theories became quietly transparent and natural
forunderstanding. Weexposedthisonmoreelementaryexamplesoftrigonometricmodelling
different motions with the use of their polar representations in quasi-, pseudo-Euclidean and
non-EuclideangeometriesandintheTheoryofRelativity. So,thehyperbolictensorofmotion
with the certain scalar multipliers produces all the kinematic and dynamic scalar, vector and
tensorphysicalrelativisticcharacteristicsofamovingmaterialbody,andgivesthegenerallaw
ofsummingmotionsandrelativisticvelocities. Thehyperbolictensorofdeformationproduces
all the relativistic seeming geometric parameters of a moving object.
Maincontentofthebookareatthejointofproblemsstudiedinmulti-dimensionalGeometry
and Linear Algebra. Since the exposition of the theory required many of additional notations
and terms, the author tried to give them the most convenient and logical forms. So, this
relatestothematrixalphabetbasedonwide-spreadingexamples. Theauthorwillbegrateful
toreaderswhowillexpressopinion,remarksorproposalsconcerningthebookonmyweb-site.
Anatoly S. Ninu´l (Dr. Ph.), December 25 2020
web-site for contacts: http://ninul-eng.narod.ru/ or
e-mails [email protected] , [email protected]
ORCID (in math): https://orcid.org/0000-0003-2861-383X
Contents
To the readers ...................................................................3
Introduction .....................................................................7
Notations .......................................................................11
Part I. Theory of Exact Matrices: some of general questions .........16
Chapter 1. Coefficients of characteristic polynomials
1.1. Simultaneous definition of scalar and matrix coefficients ....................17
1.2. The general inequality of means ............................................18
1.3. The serial method for solving an algebraic equation with real roots .........24
1.4. Structures of scalar and matrix characteristic coefficients ....................28
1.5. The minimal annulling polynomial of a matrix in its explicit form ...........34
1.6. Null-prime and null-defective singular matrices .............................37
1.7. The reduced form of characteristic coefficients ..............................40
Chapter 2. Affine (oblique) and orthogonal eigenprojectors
2.1. Affine (oblique) eigenprojectors and quasi-inverse matrix ....................43
2.2. Spectral representation of an n×n-matrix and its basic canonical form .....43
2.3. Transforming a null-prime matrix in the null-cell canonical form ............47
2.4. Null-normal singular matrices ..............................................48
2.5. Spherically orthogonal eigenprojectors and quasi-inverse matrices ...........50
Chapter 3. Main scalar invariants of singular matrices
3.1. The minorant of a matrix and its applications ..............................53
3.2. Sine characteristics of matrices .............................................57
3.3. Cosine characteristics of matrices ...........................................58
3.4. Limit methods for evaluating projectors and quasi-inverse matrices .........59
Chapter 4. Two alternative complexification variants
4.1. Comparing two variants ....................................................61
4.2. Examples of adequate complexification .....................................64
4.3. Examples of Hermitian and symbiotic complexification ......................67
CONTENTS 5
Part II. Tensor trigonometry: fundamental contents ...................68
Chapter 5. Euclidean and quasi-Euclidean tensor trigonometry
5.1. Objects of tensor trigonometry and their space relations ....................69
5.2. Projective tensor sine, cosine, and spherically orthogonal reflectors ..........71
5.3. Projective tensor secant, tangent, and affine (oblique) reflectors .............75
5.4. Comparison of two ways for defining projective tensor angles ................78
5.5. Canonical cell-forms of trigonometric functions and reflectors ...............80
√
5.6. The trigonometric theory of prime roots I ................................85
5.7. Rotational trigonometric functions and motive-type spherical angles ........87
5.8. The tensor sine, cosine, secant, and tangent of a motive type angle ..........92
5.9. Relations between projective and motive angles and functions ...............94
5.10. Deformational trigonometric functions and cross projecting ................97
5.11. Special transformations of orthogonal and oblique eigenprojectors ........100
5.12. Elementary tensor spherical trigonometric functions with frame axes ......103
Chapter 6. Pseudo-Euclidean tensor and scalar trigonometry as a basis
6.1. Hyperbolic tensor angles, trigonometric functions, and reflectors ...........107
6.2. Covariant concrete (or specific) spherical-hyperbolic analogy ...............109
6.3. The reflector tensor in quasi-Euclidean and pseudo-Euclidean interpretation 111
6.4. Scalar trigonometry in a pseudoplane ......................................116
6.5. Elementary tensor hyperbolic trigonometric functions with frame axes .....120
Chapter 7. Trigonometric interpretation of matrices commutativity
and anticommutativity
7.1. Commutativity of prime matrices ..........................................121
7.2. Anticommutativity of prime matrices pairs ................................122
Chapter 8. Trigonometric spectra and trigonometric inequalities
8.1. Trigonometric spectra of a null-prime matrix ..............................127
8.2. The general cosine inequality ..............................................129
8.3. Spectral-cell representations of tensor trigonometric functions ..............132
8.4. The general sine inequality ................................................133
Chapter 9. Geometric norms of matrix objects
9.1. Quadratic norms of matrix objects in Euclidean and quasi-Euclidean spaces 136
9.2. Absolute and relative norms ...............................................139
9.3. Geometric interpretation of particular quadratic norms ....................139
9.4. Lineors of special kinds and simplest figures formed by lineors .............141
6 CONTENTS
Chapter 10. Complexification of tensor trigonometry
10.1. Adequate complexification ................................................143
10.2. Hermitian complexification ...............................................144
10.3. Pseudoization in binary complex spaces ..................................146
Chapter 11. Tensor trigonometry of general pseudo-Euclidean spaces
11.1. Realification of complex quasi-Euclidean spaces ...........................148
11.2. The general Lorentzian group of pseudo-Euclidean rotations ..............149
11.3. Polar representation of general pseudo-Euclidean rotations ................154
11.4. Multistep hyperbolic rotations ............................................157
Chapter 12. Tensor trigonometry of Minkowski pseudo-Euclidean space
12.1. Trigonometric models for two concomitant hyperbolic geometries .........160
12.2. Rotations and deformations in elementary tensor forms ...................169
12.3. The special mathematical principle of relativity ...........................172
Appendix. TrigonometricmodelsofmotionsinSTRandnon-EuclideanGeometries.
Preface with additional notations ...............................................175
Chapter 1A. Space-time of Lagrange and space-time of Minkowski
as mathematical abstractions and physical reality .............................180
Chapter 2A. The tensor trigonometric model of Lorentzian homogeneous
principal transformations .....................................................190
Chapter 3A. Einsteinian dilation of time as a consequence of the time-arrow
hyperbolic rotation ...........................................................193
Chapter 4A. Lorentzian seeming contraction of moving object extent
as a consequence of the moving Euclidean subspace hyperbolic deformation ...197
Chapter 5A. Trigonometric models of two-step, multistep and integral
collinear motions in STR and in hyperbolic geometries ........................205
Chapter 6A. Isomorphic mapping of a pseudo-Euclidean space into time-like
and space-like quasi-Euclidean ones, Beltrami pseudosphere ...................222
Chapter 7A. Trigonometric models of two-step, multistep and integral
non-collinear motions in STR and in hyperbolic geometries ...................228
Chapter 8A. Trigonometric models of two-step and multistep non-collinear
motions in quasi-Euclidean space and in spherical geometry ...................260
Chapter 9A. Real and observable space-time in the general relativity ..........272
Chapter 10A. Motions along world lines in (cid:104)P3+1(cid:105) and their geometry ........285
Mathematical–Physical Kunstkammer ..........................................304
Literature ..................................................................... 308
Name Index ...................................................................312
Subject Index ..................................................................315
Abstract .......................................................................319
Exit data ......................................................................320
Introduction
InTheoryofMatricessuchusualnotionsasasingularmatrix,itsrank,eigenvalues,
eigenvectors or eigensubspaces, annuling polynomial, and so on, have a sense only for
exact matrices and at exact computations. We distinguish the exact theory of notions
and the approximating theory of notions estimates. Each of them places its own part.
Thenotionsconnectedwithexactnumericalcharacteristicsrelatetotheexacttheories.
Thesetheoriesareusednotonlyforconstructingandanalysisofabstractions,butthey
arealsoimportantforanalyzableobjectsfromappliedproblemsbecausethenumerical
characteristics of objects are always exact and only their estimates are approximate.
Themaintwopartsofthemonograph,intwelvechapters,containboththeresultsof
ourinvestigationsinTheoryofExactMatrices(PartI,chapters1÷4)anddevelopedon
thisplatformTensorTrigonometry(PartII,chapters5÷12). Thelatterisaconstituent
division of the corresponding to it Geometry with a certain quadratic metric.
ThehistoricalrootsofScalarTrigonometry,asaconstituentpartoftwo-dimensional
Geometry,refertofar-awaytimes. YetintheEuclidean”Elements” sometrigonometric
formulations were be found. Much later, in II age Claudius Ptolemy of Alexandria
widely used in "Almagest" sine-cosine formulae with his trigonometric equivalent of
the Pythagorean Theorem. Some spherical functions were used also in IX–X ages
in the works of Arabian mathematicians. It is of interest that the Trigonometry on a
spherebecamedevelopedmuchearlierthantheoneonaplane. Itwas,duetothefact,
that it was needed in the practical astronomy. So, in 1603, Th. Harriot connected the
angularexcessofasphericaltrianglewithareaandradius. Thoughsometrigonometric
elementswereintroducedintotheEuropeansciencebyR.Wallingfordinthebeginning
of XIY age. Thus, in particular, he used it in solving of a right triangle.
HyperbolicfunctionswerediscoveredbyA.Moivre(1722)andobtainedincomplete
setbyV.Riccatifromaunityhyperbola(1757). Firstthesefunctionswereusedreally
in hyperbolic Scalar Trigonometry and in geometric investigations by J. Lambert and
F. Taurinus. So, in 1763, J. Lambert connected the angular defect of a hyperbolic
trianglewithareaandradius. In1825,F.Taurinusdiscoveredthefirst(cosine)formula
forsummingtwosegmentsinthehyperbolicgeometry. Creatorsofthehyperbolicnon-
Euclidean geometry N. Lobachevsky and J. Bolyai used it analogy in the small with
the spherical geometry as a main mathematical instrument for inferences of its metric
formulae. These geometries have such distinction: their geodesic arcs–segments are
hyperbolicandspherical. Inpseudo-Euclideanorsocalledquasi-Euclideangeometries,
there are straight segments, but with hyperbolic or spherical angles between them!
The modern perfect form of the Scalar Trigonometry was given by L. Euler, who
realized also its complexification. On the other hand, Geometry continues to develop
and essentially violently according to the appeared idea of a multi-dimensional space.
8 INTRODUCTION
Multi-dimensionalspacewasarisenapparentlyatthemiddleofXIXageinclassical
workofH.Grassmann”DielinealeAusdehnungslehre” [1]. H.Grassmannand,indepen-
dently of him, W. Hamilton laid the foundation of Vector Analysis in the spaces.
Before (in 1808) J.-G. Garnier emits Analytical Geometry as the whole division of
Geometry. The outstanding contribution in justification of the algebraic approach to
the geometry of any objects in multi-dimensional arithmetic spaces was realized by
the famous ”Cantor–Dedekind Axiom about Continuum”.
About of that time appearance of Linear Algebra and its following development in
the works of F. Frobenius, G. Cramer, L. Kronecker, A. Capelli, J. Silvester, L. Hesse,
C. Jordan, Ch. Hermite and other mathematicians led, with time, to its larger filling
by geometric content. That is why, Linear Algebra found effective applications in the
theory of vector Euclidean spaces and also, after the well-known works of H. Poincare
and H. Minkowski, in the theory of new pseudo-Euclidean spaces. This process was
activatedthankstoalgebraicdefinitionsofnotionsconnectedwithmetricpropertiesof
arithmetic spaces and of their geometric objects (the lengths of vectors and the values
of scalar angles between them). As well-known for the basic algebraic definitions of
measures mathematicians used the Pythagorean Theorem and the algebraic cosine
Inequality of Cauchy or sine Inequality of Hadamard.
Besides, for the strict algebraic approach to the geometry in arithmetic spaces, it
is impossible to realize it completely without Theory of Exact Matrices. For example,
E. Moore and later R. Penrose proposed the general methods of quasi-inversion of
singular matrices. R. Courant developed the large parameter optimization method
with penalty functions, useful in such algebraic applications too. A. Tichonov gave
thesmallparametermethodofregularizationwiththelimitmethodfornormalsolving
degenerated systems of linear equations. Results of these investigations had also a big
geometric importance and, to some degree, served for initiating the present work.
Themainaimsofthismonographwere(as1st)todevelopwithfurtherapplications
a number of algebraic and geometric notions in Theory of Exact Matrices (Part I,
chapters 1÷4), and then (as 2nd) on the platform to work out the basic aspects
of Tensor Trigonometry for binary tensor angles formed by two linear subspaces or
formed by rotation of a linear subspace in the superspace (Part II, chapters 5÷12).
Since Tensor Trigonometry has a lot of applications in other mathematical and some
physical domains, the largest examples of which are exposed in the book’s Appendix.
First of all, the structure of matrix characteristic coefficients for n×n-matrices in
the explicit form is installed by special differential method. They appeared in Theory
ofExactMatricesinmiddleofXXageintheworksofJ.-M.SouriauandD.K.Faddeev
in addition to scalar characteristic coefficients with the well-known structure. These
lastwereusedyetinXIXagebyU.LeVerrierathisfamouspredictionofNeptune. We
express all eigenprojectors and quasi-inverse matrices in explicit form, in terms of the
scalarandmatrixcoefficients. Andtheminimalannullingpolynomialforn×n-matrix
in explicit form is identified with the connections of all matrix singularity parameters.
INTRODUCTION 9
In passing, the general inequality for all average values is inferred, and hierarchical
invariants for the spectrally positive matrix are installed for the justification of the
stated geometric norms. The new global limit method for step by step calculating all
roots of a real algebraic equation is proposed, and the more strict necessary condition
for all its roots reality and positivity, than the classical Descartes condition, is gotten.
The particular (of order t) and general (of order r) quadratic norms (measures) are
introduced for the linear geometric objects lineors, determined by n×r-matrices A,
where 1 ≤ r < n (at r = 1 they are vectors), and for the tensor angles between them
or between their images in the n-dimensional arithmetic spaces. In particular, at t
= 1 they are Euclidean and Frobenius norms (measures). The theoretical basis for
theseparticularandgeneralnormsisthehierarchicalgeneralinequalityforallaverage
positive values. Also the specific multiplications of cosine and sine types are defined
forapairoftheselineorswithinferringthegeneralcosineandsineinequalitiesthrough
thespecialmatrixtrigonometricspectra. Theirelementaryalgebraicandtrigonometric
cases are the cosine Inequality of Cauchy and the sine Inequality of Hadamard.
Tensor Trigonometry, as the main new content of this monograph, is exposed then
in its three kinds: projective, reflective and motive, which naturally complement each
others. Two types of motive trigonometric transformations, as rotational (sine-cosine)
and deformational (tangent-secant), are defined. Besides, the general homogeneous
transformations,duetotheformerpolarrepresentations,aredividedaseitherprincipal
sphericalandorthosphericalrotationsorprincipalhyperbolicandorthosphericalones.
Thespecialdualrelationsareestablishedbetweenprimarysphericalandhyperbolic
notions on the basis of spherical-hyperbolic analogy in abstract and specific senses.
This is widely used in developing Tensor Trigonometry and in its applications. So,
similar definitions of quasi-Euclidean and pseudo-Euclidean metric spaces and their
Tensor Trigonometries are exposed, in terms of the initially given reflector tensor of a
binary space with the kind of quadratic metrics (Euclidean or pseudo-Euclidean).
In the pairs (spherical, orthospherical), (hyperbolic, orthospherical) all rotations in
thesebinaryspaces,withanidenticalreflectortensor,formtwononcommutativegroups.
Thefirstoneisthehomogeneousgroupofquasi-Euclideanmotions. Thesecondoneis
the homogeneous pseudo-Euclidean group of Lorentz. In the so-called universal bases,
the intersection of these two groups is a proper subgroup of secondary orthospherical
rotations. This reflector tensor and both quadratic metrics define in the same spaces
two sets of reflective transformations. Obviously, reflections do not form groups.
Mostimportantly,thequasi-Euclideanspaceandgeometryfilledinthismonograph
apreviouslyunnoticedgapthatexistedinthetheoryofhomogeneousisotropicspaces.
This can be explained by the fact that the pseudo Euclidean space with the Lorentz
group was introduced back in 1905 by H. Poincar´e as the mathematical apparatus of
the theory of relativity. This name was given to it later by M. Planck. The geometry
of this space was developed by H. Minkowski in 1907. But mathematically, they are
pseudoanalogues namely of this original quasi-Euclidean space and its geometry!