Elementary Matrices And Some Applications To Dynamics And Differential Equations

432 Pages·1952·15.13 MB·English

by  A. R. Collar

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ELEMENTARY
MATRICES

ReATRAZER
‘Wi. f-UNCAN
AR COMLAK
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PRESTON POLYTECHNIC
LIBRARY f LEARIING RESOURCES SERVICE
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30107 533 924
ELEMENTARY MATRICES
AND SOME APPLICATIONS TO DYNAMICS
AND DIFFERENTIAL EQUATIONS

R.A. FRAZER, B.A., DSC., FR.AES., FLAZS., ERS.
"Pc ic On he Aram ion
“thas pea Lao
W. 1, DUNCAN, D.Sc, MIMECKE., FRARS.
Dofnar of stam Coke ros, Celt

ALR. COLLAR, M.A., DSc, FR.ABS.

Sir Gorge Whee Prafenrof erent Engen
Tse Unberaty of ri

CAMBRIDGE: AT THE UNIVERSITY PRESS
1952
Zooey
CONTENTS

Brice
CHAPTER I
BUNDAMENTAL DEFINITIONS AND
;: ELEMENTARY PROPERTIES

14. Preliminary Romecks

12. Notetion anc Prinsgal Type of Matrix

13 Summation of Marios and Salar Moltpies

14 Multipiction of Mateioxs

145 Cnntnued Produsta of Motions

116 Propertie of Diagonal end Ualt Matrons

U7. titioning of Matrise into Submatsioes

48 Determinants of Square Masioss

19 Singular Matrices, Degonsresy,ané Rani

140 Adjeint Matrices

LAI Reciprocal Matsieee nd Division

1142 Square Matrions with Noll Product

1143 Reversal of Ontr in Prone when Mateose are Teanspoeod
or Reoiprocstad

114 Linear Substications

4145 Biliauas aust Quaseatio Poems

1146 Discriminants snd One Signed Quedrutie Forms

{LAT Speco) Type of Square Matrix

oHAPTER

POWERS OF MATRICES, SERIES, AND
INEISITESIMAL CALCULUS

24 Intiodatory

22 Power of Matriows

263. Rolymomine of asi

2-4 Infinit Serie of Matricos

25 The Expenential Funotion

2.6 Difisentintion of Matrace

BeREe

Gess2s
an.

27° Differentiation ofthe Exponential Function
28 Mateiws of Diferential Operators

29 Chango ofthe Independant Variablon

2.10 Integration of Matvious

2.1 Tho Mateaant

onaPrER ut
LAMBDA.MATRICES AND CANONICAL FORMS
34 Prolininary Romaske
Paw L Lambla-Matices
32 Lambda Mattoes
33 Multiplication and Division of Lambda Matrices

34 Remainder Theorems for Lamide-Matrios

35 ‘Tho Dotorminantal Byustion and the Adjoit of « Lambda
‘Matrix

36 ‘The Charactcisto Matrix of « Square Mats and the Tatent
Roots

3.7 The Capley-Hansilton Theorem,
3.8 Tho Adjoiot and Derived Adjint ofthe Chasnctersto Mate:
3.9. Sylvester's Theowom

‘3:40 Content Form of Bptvostr’s Tore

Par 11. Canonical Forme

3.11 Klomentary Operations on Matrioos
3:12 Rouivalent Matrons

13.13 A Canoniel Fora Jor Square Matios of Rank r
3.44 Kauivalont Lambda-Mateioos

‘3-18 Saith’s Canonical Forn for Lambda Matrices

3:16 Colincstory Transformation of Numerical Matrix to a
‘Ganonia! Form,
conrunrs

cuarren iv

jg, MISCELLANEOUS NCMBRICAL MaTHODS

4:1” Range of tho Subjesta Trostod

ane L, Determinants, Roiprocal aud Aijoins Matrices,
‘and Systoma of Linccr Algebraic Batons

42, Protiminary Remasis

42. Triangular and Related Mateoes

44, Redaction of Tangalne ond Related Mateooe to Diagonal
‘Form

45° Reciproras of Tiangular and Melted Matioos

45 Computation of Determinants

47 Computation of Resiprosal steios

48. Recipocation by the Methed of Postmaltipliors

49° Reciprooation by the Methed of Submatrion

4410 Reciprosaton by Diet Operations om Rerws

4411 Improvement ofthe Accarsoy of an Appeoximata Resirosal
‘Mois

412 Computation ofthe Ajint ofa Singular Matric

1413 Numerical Sclution of Simultansoue Linear Algebraic Equa.
‘ene

Pane UL, High Powero of ¢ Matrix andthe Latent Rots

4.14 Proiminacy Sommary of Svusters Theerom

45 Eralaston ofthe Dominant Latent Roots from the Limiting
"Perm of @ High Power of » Matric

416 Rraluation of the Matsix Coeficiente 2 for tho Dominant
Hoots

4.17 Simple Iterotive Methods

{418 Compitation of the Non-Dominant Latent Roots

4.19 Upper Bounds t the Powers of Matrix

Part UL, Algetrnie Rguations of General Deyres

1420 Soinion of Algetnso Ruane and Adaptation of Alison's
ermal

421 Conc! Ksiarka on Tenatve Mathods

422 Situation of the Root ofan Algebraic Bquation

a

2

ast
OMAPTER V

LINZAR ORDINARY DIFFERENTIAL EQUATIONS

Pane I, Geterat Propertoe

ar.
454 Syetems of Simultencous Differential Equations
152. Equivalent Systeme

45. ‘Transformation ofthe Dependons Variable

54 ‘Triangular Syetems and Purdeduitel Torun

55 Conversion ofa System of General Onder into 8 Bist-Ondor
‘System

5.6 Tho Adjoint end Derive Adjoine Matrices

5:7 Construction of the Constituent Solations

5.8. Numerisal Rvaluation of the Constituent Sot

5.9 Expansions in Partial Fractine

‘Pane IL. Construction of the Complementory Fenetion
‘and of Pariodar Intgrad

5:10 Tho Compicmontary Function

5.11 Construction of » Paonia: Intgral

LINEAR ORDINARY DIFFERENTIAL EQUATIONS
WITH CONSTANT COEFFICIENTS (cominuct

PaweT. Bowndary Prollene

61. Protininary Remarie

62. Charcteritie Numbers

63 Notation for Ono-Point Boundary Problems

64 Dirovt Solution of the Genceal Oue-Pont Boundary Preblem

65 Special Solution fr Starsand O-Point Houmas Pecleune

66 Content Fon of the Syeda Slaton

67 Notation ond Dinect Solation for ‘Two-Puiat Doundery
Problems

15

18
188
s, ‘Pana IL. Stems of Firat Onder

68. Proliminary Remaehs
(69) Spesial Solution of the General Fist- Onder System, ond ite
“Connection with Hearse’ tho

610 Determinantel Kguation, Adjoizt Msteces, and Mod
Colum forthe Simple First-Order Syatem

6-1 Generel, Die, and Special Solokions of the Simple First.
‘Order Bystom

{612 Power vies Saluson of ple Fest Onler Sytem

418 Power Beciee Salton of to Simple First-Order Sytem for 8
"Two-Point Boundary Proton

NUMPRICAL SOLUTIONS OF LINEAR ORDINARY
(DIFFERENTIAL EQUATIONS WITH
VARIABLE COEFFIOVENTS

7A Range ofthe Chapter

172. Bristenos Theor and Bingularsios

73 Fundamental Solutions of Single Linear Homogeneous
Begetion

7-4 Syms of Santarsoue Lizese Difrontial Equations

75 Tae Peano-Baksr Method of Inayration

7-6 Various Propctes ofthe Matrzant

77 A Continuation Formals

178 Solution of the onnogencons Vind Oxdae Systm of Bquasions
‘in Power Sarcs

179 Colleton ad Galerkin’ Method

17410 Bxamplos of Numerical olusion hy Cllocation sud Golesi
Method

1744 he Method of Mens Cooiants
17.42 Golution by Most Coolie: Hamp No.1
‘743 Maple No.2
7.44 Bxample No. 3
745 Example No. 4
x contents

OMAPTER VIIT
KINEMATICS AND DYNAMICS OF SYSTEMS

Pant I, Frames of Reference and Kinematics

ant a
84 Frame of Refornce 286
182 Chango of Reference Axes in Two Dimensions 2st
8.3 Angular Coonlinntes of « Thrve-Dinensinal Moving Frame
of Reference 250
‘The Orthogonal Matix of Traneermation 251
“Matrioes Reproxnting Vinte Rotation of « Frame of Rafer.
nae 25
8.46 Matrix of Transformation and Tastantancous Angular Valo
sites Bxpresed in Angus Conrdinstce 258
8.7 Components of Velocity and Aselertion 258
88 Kinematic Consteaint of a Rigid Body 250
8.9 Syston of Rig Bodin and Genrrlied Cocrdnntae 200
Pauw TI. Statice and Dynamics of Systane
{8.10 Virtual Work and the Conditions of Equilibrium 262
£8.11 Conservative and Non-Conservative Fields of Fores 263
8.12 Dynamical Systems 206
8.13 Bquesions of Motion of an Aeroplane 267
8.14 Lagrange's Bountions of Motion of a Holonomoxs Syatom 200
£8.48 Ignoration of Coordinates mm
8.16 The Gonerliod Componente of Momentom and Hamion'
Byuations mm
8.17 Lagrange’ Bqutions with s Moving Frame of Refrense 277

SYSTEMS WITH LINBAR DYNAMICAL EQUATIONS

9.4 Tntroduetory Remarks 280
92 Diturbed Motions 280
9:9 Comervative Systom Disturlol fous Bquilrians 231
9-4 Disturbed Stendy Motion of m Consrvativo System with
‘gnorable Coordinates 22