Linear Algebra and Its Applications

394 Pages·2007·3.687 MB·English

by  Lax, Peter D

Linear Algebra and Its
Applications
pPISSeTenmiar
«1=| 148073
1 'z
THE WILEY BICENTENNIAL-KNOWLEDGE FOR GENERATIONS

ach generation has its unique necds and aspirations. When Charles Wiley first
opened his small printing shop in lower Manhattan in 1807, it was a generation
of boundless potential searching for an identity. And we were there, helping to
define a new American literary tradition. Over half a century later, in the midst
of the Second Industrial Revolution, it was a generation focused on building the
future. Once again, we were there, supplying the critical scientific, technical, and
engineering knowledge that helped frame the world. Throughout the 20th
Century, and into the new millennium, nations began to reach out beyond their
own borders and a new international community was born. Wiley was there,
expanding its operations around the world to enable a global exchange of ideas,
opinions, and know-how.
For 200 years, Wiley has been an integral pact of each generation’s journey,
enabling the flow of information and understanding necessary to meet their needs
and fulfill their aspirations. Today, bold new technologies are changing the way
we live and learn. Wiley will be there, providing you the must-have knowledge
you need to imagine new worlds, new possibilities, and new opportunities.
Generations come and go, but you can always count on Wiley to provide you the
knowledge you need, when and where you need it!
Weert, Bee (ELK: tll
WILLIAM 2. PESCE PETER BOOTH WILEY
PRESIDENT AND Crier EXECUTIVE OFFICER CHAIRMAN OF THE BaARD
Linear Algebra and Its
Applications
Second Edition
PETER D. LAX
New York Universily
Courant Institute of Mathematical Sciences
New York, NY
BICENTENNIAL

L S
A I
I S
N Z
N N
ET AZ
N N
E N
C I
I V
B A
BICENTENNIAL
WILEY-INTERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2007 by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved.
Published sinwultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as
permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the
prior written permission of the Publisher, or authorization through payment of the appropriate
percopy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923,
(978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed
to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012,
(212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM.
For ordering and customer service, call 1-800-CALL-WILEY.
Wiley Bicentennial Logo: Richard J. Pacifico
Library of Congress Cataloging-in-Publication Data:
Lax, Peter D.
Linear algebra and its applications / Peter D. Lax. — 2nd ed.
p. cm. — (Pure and applied mathematics. A Wiley-Interscience of texts, monographs and tracts)
Previous ed.: Linear algebra. New York : Wiley, c1997.
Includes bibliographical references and index.
ISBN 978-0-471-75156-4 (cloth)
1. Algebras, Linear. I. Lax, Peter D. Linear algebra. II. Title.
QA184.2.L38 2008
512°.5—dce22
2007023226
Printed in the United States of America
10987654321
C o n t e n t s
P r e f a c e A l
P r e f at oct eh eF i r sEtd i t i o n K i l i
F u n d a m e n t a l s 1
L i n e aS rp a c eI,s o m o r p h i s m 1
S u b s p a c e 4
L i n e aD re p e n d e n c e 4
B a s i Ds ,i m e n s i o n 5
Q u o t i eS np ta c e 8
D u a l i t y 1 3
L i n eFaur n c t i o n s 1 3
D u a lo f a L i n e aS rp a c e 1 4
A n n i h i l a t o r 1 5
C o d i m e n s i o n 1 6
Q u a d r a Ft ou rr em u l a 1 7
L i n e Ma ra p p i n g s 1 9
D o m a ainndT a r g eS tp a c e 1 9
N u l l s paancdRe a n g e 2 0
F u n d a m Te hn te ao lr e m 2 0
U n d e r d e t eLrimn ie Sna rye sd t e m s 2 1
I n t e r p o l a t i o n 2 2
D i f f e r eEnqcuea t i o n s 2 3
A l g e borf L ai n e Ma ra p p i n g s 2 4
D i m e n soif N o unl l s paancdRe a n g e 2 7
T r a n s p o s i t i o n 2 6
S i m i l a r i t y 2 9
P r o j e c t i o n s 3 0
M a t r i c e s 3 2
R o wasn dC o l u m n s 3 3
M a t r iM xu l t i p l i c a t i o n 3 5
V I C O N T E N T S
T r a n s p o s i t i o n 3 6
R a n k 3 7
G a u s sEilainm i n a t i o n 3 9
D e t e r m ai nn dTa rn at c e 4 4
O r d e rSeidm p l i c e s 4 4
S i g n eVdo l u mDee,t e r m i n a n t 4 5
P e r m u t aGtrioounp 4 6
F o r m uf olrDa e t e r m i n a n t 4 8
M u l t i p l i cP ar to ip veer t y 4 9
L a p l aEcxep a n s i o n 5 2
C r a m eRru’lse 5 4
T r a c e 5 5
S p e c t rTahle o r y 5 8
I t e r a toifo Lni n e aM ra p s 5 9
E i g e n v a lEuiegse,n v e c t o r s 5 9
F i b o n aScec qi u e n c e 6 2
C h a r a c t e rPiosltyinco m i a l 6 3
T r a caen dD e t e r m Ri en vainsti t e d 6 5
S p e c t rMaalp p iTnhge o r e m 6 6
C a y l e y — H aTmhielotroenm 6 7
G e n e r a lEiizgeedn v e c t o r s 6 9
S p e c t rTahle o r e m 7 0
M i n i mPaoll y n o m i a l 7 2
W h e Anr eT w oM a t r iSciemsi l a r 7 3
C o m m u tMianpgs 7 4
E u c l i dS et ar nu c t u r e 7 7
S c a lParro d uDcit s, t a n c e 7 9
S c h w aIrnge q u a l i t y 7 9
O r t h o n oB ar smiasl 8 0
G r a m — S c h m i d t 8 1
O r t h o g oCnoamlp l e m e n t 8 2
O r t h o g oPnr ao lj e c t i o n 8 3
A d j o i n t 8 4
O v e r d e t e rS my is tn ee md s 8 6
i s o m e t r y 8 7
T h eO r t h o g oG nr aolu p 8 9
N o r omf a L i n e Ma ra p 8 9
C o m p l e t eL no ce asClso m p a c t n e s s 9 2
C o m p lEeuxc l i dSetarnu c t u r e 9 5
S p e c t rRaald i u s 9 7
H i l b e r t - SN co hrmmi d t 9 8
C r o sPsr o d u c t 9 9
C O N T E N T S Vil
8. SpectraTlh e o r oyf Self-AdjoMiantp p i n g s 101
Q u a d r a tFiocr m s 102
L a wo f Inertia 103
SpectraRle solution 105
C o m m u t iMnagp s 111
Anti-Self-AdjMoaipnst 112
N o r m a lM a p s 112
RayleigQhu otient 114
M i n m aPxr inciple 116
N o r ma nd Eigenvalues 119
C a l c u loufs V e c t o ra-n dM a t r i x - V aFluunecdt i o n s 121
C o n v e r g e ninc Neo r m 121
R u l e sof D i f f e r e n t i a t i o n 122
Derivatiovfe d et A(f) 126
MatrixE xponential! 128
SimpleE igenvalues 129
MultiplEei genvalues 135
R e l l i c hT’hse o r e m 140
Avoidancofe C rossing 140
10 M a t r i Ixn equalities 143
PositivSee lf-AdjoMianttr ices 143
M o n o t oMnaet r iFxu n c t i o n s 151
G r a mM a t r i c e s 152
S c h u r ’Tsh e o r e m 153
T h eD e t e r m i noaf nP ots i t iMv ea t r i c e s 154
IntegraFlo r m u lfaor D e t e r m i n a n t s 157
Eigenvalues 160
Separatioofn E igenvalues 161
W i e l a n d t ~ H oTfhfemoarne m 164
Smallesatn d LargesEti genvalue 166
Matricewsi thP ositivSee lf-AdjoPianrtt 167
PolarD e c o m p o s i t i o n 169
SingulaVra lues 170
SingulaVra lueD e c o m p o s i t i o n 170
I] K i n e m a t iacnsd D y n a m i c s 172
Axisa nd Angleo f Rotation 172
RigidM o t i o n 173
A n g u l aVre locitVye ctor 176
F l u i dF l o w 177
Curla nd D i v e r g e n c e 179
S m a l lV i b r a t i o n s 180
Conservatoifo Enn e r g y 182
Frequenciaends N o r m a Ml o d e s 1 8 4
Vill C O N T E N T S
12. C o n v e x i t y 137
C o n v eSxe ts 187
G a u g eF unction 188
H a h n — B a nTahceho r e m 191
S u p p o rFtu n c t i o n 193
CarathéodorTyh’eso r e m 195
K O n i g — B i r kThho fefo r e m 198
Helly’Tsh e o r e m 199
13 T h e DualitTyh e o r e m 202
F a r k a s — M i n kTohwesokr1e m 203
DualitTyh e o r e m 206
E c o n o m iIcnst erpretation 208
M i n m aTxh e o r e m 210
14 N o r m eLdi n e aSr p a c e s 214
N o r m 214
l? N o r m s 215
Equivalenofc eN o r m s 217
C o m p l e t e n e s s 219
LocalC o m p a c t n e s s 219
T h e o r eofm F . R i e s z 219
D u a lN o r m 222
Distancfer omS u b s p a c e 223
N o r m eQdu otienStp ace 224
C o m p l eNxo r m e Sdp aces 226
C o m p l eHxa h n - B a n aTchhe o r e m 226
Characterizaotf iEounc lideSapna ces 227
Is LinearM a p p i n gBse t w e eNno r m eLdi n e aSrp a c e s 229
N o r mo f a M a p p i n g 230
N o r mo f T r a n s p o s e 231
N o r m eAdl gebroaf M a p s 232
InvertibMlae p s 233
SpectraRla dius 236
16 P o s i t iMv ea t r i c e s 237
P e r r o nT’ sh e o r e m 237
S t o c h a s tMiact r i c e s 240
F r o b e n i uTsh’e o r e m 243
17 H o w t o SolveS y s t e mosf L i n e aEr q u a t i o n s 246
History 246
C o n d i t iNounm b e r 248
IteratiMvee t h o d s 248
SteepesDte scent 249
C h e b y c hIetve ration 2 5 2
C O N T E N T S
Three-teCrhme b y c hIetve ration 255
OptimaTlh r e e - T eRremc ursioRenl ation 256
Rateo f C o n v e r g e n c e 261
18 H o w to Calculatthee E i g e n v a l uofe Sse lf-AdjoMianttr i c e s 262
O R Factorization 262
Usingt he Q R Factorizattio oSno lveS y s t e mosf Equations 263
The Q R A l g o r i tfhomr FindinEgi g e n v a l u e s 263
H o u s e h o lRdeefri ectifoonr Q R Factorization 266
TridiagonFaolr m 267
A n a l o goyf Q R A l g o r i tahnmd T o d aF l o w 269
M o s e r ’Tsh e o r e m 273
M o r eG e n e r aFll o w s 276
19. S o l u t i o n s 278
B i b l i o g r a p h y 300
A p p e n d i1.x SpeciaDle t e r m i n a n t s 302
A p p e n d i2.x T h eP f a f f i a n 305
A p p e n d i3.x S y m p l e c tMiact r i c e s 308
A p p e n d i4.x T e n s oPrr o d u c t 313
A p p e n d i5.x L a t t i c e s 317
A p p e n d i6.x FastM a t r iMxu l t i p l i c a t i o n 320
A p p e n d i7.x G e r s h g o r iTnh’ se o r e m 323
A p p e n d i8.x T h e Multiplicoift Eyi g e n v a l u e s 325
A p p e n d i9.x T h eF a s tF o u r i eTrr a n s f o r m 328
A p p e n d i10x. The SpectraRla d i u s 334
A p p e n d i11x. The L o r e n tGzr o u p 342
A p p e n d i12x. C o m p a c t n eofs tsh e UnitB all 352
A p p e n d i13x. A C h a r a c t e r i z aotf iCoonm m u t a t o r s 355
A p p e n d i14x. L i a p u n o vT’hse o r e m 357
A p p e n d i15x. The J o r d aCna n o n i cFa lo r m 363
A p p e n d i16x. N u m e r i cRaaln g e 367
I n d e x 3 7 3