bronson, r schaum's outline of theory and problems of matrix operations

Moliply the clemenlS of any r0 by 2 nonzero scalar £3; Add to any row, element by clement, 2 scalar times the corresponding elements of another “Three elementary column operations are

SCHAUM'S OUTLINE OF THEORY AND PROBLEMS

MATRIX OPERATIONS

RICHARD BRONSON, Ph.D

Profestor of Mathematics and Computer Science

"Faireigh Dickinson University

SCHAUM’S OUTLINE SERIES

McGRAW-HILL, [New York San Francisco Washington, D.C Auckland Bogots Coracus Lisbon

London Madrid Mexico City Milan Montrel New Dehli

Sen luan Singapore Sydney Tokyo Toronto

To Evy

RICHARD BRONSON, whois Professor and Chairman of Mathematics 2d Computer Science at Fasleigh Dickinson University, received Bis Ph.D in applied mathematics fom Stevens Institute of Technology in

1968, Dr Bronson is currently an associate editor of the journal Simula-

‘ion, contibuting editor to STAM News, has served asa consultant to Bell

‘Telephone Laboratories, and has published over 25 technical articles and

‘Schaom's Outie of Theory and Prins of

‘Copyright© 1989 by The MeGraw-il Compares Ine Al ight reseed Prien be

‘nied Ses of Ameria Except pemined wer he United Str Copyiht Ac of

1976, no par of his publication may e reproduced odie aw fom ob 3

‘memo sored ins dts eo rina em, io he pe wten perm of

(Seman aati ses)

P'S Te, Tile: Mae

Preface

Pethaps no area of mathematics has changed as dramatically as matrices over the last 25 years This ir due to both the advent ofthe computer as well asthe introduction and acceptance of matrix methode into other applied discipline: Computers provide an efficient mechanism for dong erative computations This,

in turn, has revolutionized the methods wied for locating eigenvalues and eigenvectors and has altered the wefulness of many classical techniques, such 28 those for obtaining inverses and solving simultaneous equations Relatively new felis, such as operations research, lean heavily on matrix algebra, while estab- lished fields, such as economics, probability, and differential equations, continue

to expand their reliance on matrices for canlying and simplifying complex

‘concepts

‘This book is an algorithmic approach to matrix operations The more complicated procedures are piven as 8 series of steps which may be coded in 3 straightforward manner for computer implementation The emphasis throughout {son computationally efficient methods These shouldbe of value to anyone who

‘needs to apply matrix methods to his or her own work

‘The material inthis book is selfcontained; all concepts and procedures are stated directly in terms of matrix operations There are no prerequisites for using

‘most of this Book other than a wotking knowiedge of high school algebra Some

Of the applications, however, do requte additonal expertise, but these are selfevident and are limited t0 short portions ofthe book For example, elemen tary calcuhs i needed forthe materal on differential equations

Each chapter of this book is divided into thee sections The fis introduces concepts and methodology The second section consists of completely worked-out [problems which clarify the material preseated inthe ist section and which, on

‘ceasion, leo expand on that development Finally, there i section of problems

‘with answers with which the reader can test his or her mastery of the subject

| wish fo thank the many individuals who helped make this book a realty 1

‘warmly acknowledge the coatsbutions of Wiliam Anderson, whose comments on

‘overage and content were particularly valuable I'am also grateful to Howard

‘Karp and Martha Kingsley for their suggestions and assistance Particular thanks are due Edward Millman for his splendid editing and support, David Beckwith of the Schaum staf for overseeing the entire project, and Marthe Grice fr technical editing

‘Maures Vecors and dot progecs Matt aditon and mati isbrscion Scalar nlipicatoe nd mast mulipicaon.” Row echelon form Elementary row and cole operations Rank

Consistency Matix notation Thewy gfmlelots Simpiing operons Gaunian eiminatonalgontha, Photing statepes

Đigonak, Elenemmy maưees LU đeeompowion Sinitancous linear equations Powers of «matrix

Themhene, Simgi imesex, Cacelsingmwenek Smalaaese laser equations Properties of the verse

Expansion by ofcton_ Properties of determines Determinants of pariioned matncer Pott condenunon Tatenlon by determinants

Dimension Linear dependence and independence Lineat combinations

Propet of inary dependent vectors Row rank ad column rank

Charscerisic equation Properties of eigenvalus and eigenvectors Leary Independent eigenvector Computational considerations The Capey Hamsiton theres

Sequences and sens of matrices Welldeined functions Compatng functions of matnes The function <*”Daferentition and iteration of imatiees Difleental equations The matnx equation AX XS C,

Generales eigenvector Chains Canonical basi The minum peor

‘Similar matrices Modal mtr Jorn canonical form, Siiity and

Jordan canonical form Functions of maces,

Complex conjugntes The ser prodect Properties of inner products Onhogonaiy Gram Schauer orbogonazston

Vector norms” Normaited vectors and dtance Mati norms Indsced mưme ComplblA) Spec reo

[Normal matices, Hermitian sasices Real symmetric mrices.” The dj" Selfajoint mares

Delite maces Tests for psive deinteness Square root of smatices Cholesky decomposion

Unitary matices Schur decomposition Elementary refecion Summary of simian transformations

(usdrae form Diagonal form Congruence Inertia Rayleigh quotient,

Nonnepaive and postive mainces reduce mnces Prine aimee Stochatle mattces inte Markov cha

‘Grealat matices Band matrices Thiagonal matrices Hessenberg form

Chapter 19 POWER METHODS FOR LOCATING REAL EIGENVALUES

Numerical methods, The power metho The inverse power method The

‘hited inere power method Genchgorns theorem Chapter 20 THE QR ALGORITHM

‘The modifies Gram-Schmist proces QM decomposition The QR algorithm Accelerating convergence

Chapter 21 GENERALIZED INVERSES

Properties A formula fr generalized iveses_Sigulrvalue

<ecompostion, A stable formala forthe generalized inverse Least-squares

ANSWERS TO SUPPLEMENTARY PROBLEMS,

INDEX

i

19

4 complex-valued function, If alls elements are numbers, then a matrix s called & consi! mattix

Example 1

ae all matices The fret to onthe eft ace re-valve, wheres the hid i complexed (wih = V=T)s the st and tied are constant mates, but the second not constant

Matrices are designated by boldface uppercase letters A general matrix A having r ows and c columas may be written

A-|#t 22.2

where the elements ofthe matrix are double subscripted to denote location, By convention, the row index precedes the columa index, thus, 2, eprcxent the element of A appearing inthe second fow And fh columa, while ay, represents the element appearing in the thd row and fst columa A tmatrix A may also be denoted a5 [e,), where a, denotes the general element of A appearing in the {th row and jth column

'A matrix having 7 fows and c columas has order (o size) “r by ¢." usually written v xc, The matrices ia Example Ll have order 2%2, 23, and 14, respectively from left to right Two matrices are equal they have the same order and their cortesponding elements are equal

“The Eengeose of a matix A, denoted as A’, is obtained by converting the rows of A into the columns of AY ope ata time in sequence If A has order mn, then A has order mm

Example 1216

“B=

VECTORS AND DOT PRODUCTS

{A vector i a matrix having either one row or one column A matrix consisting of singe ro called a row vecior: a matrix having a single column i called a column vector The dot product A> B fof two vectors ofthe same order is obtained by multiplying topether corresponding elements of A and Band then summing the results The Jot produc isa scalar by which we meaa sis ofthe same

‘eneral type asthe elements themselves (See Problem 1.1.)

2 BASIC OPERATIONS [onan 1

[MATRIX ADDITION AND MATRIX SUBTRACTION

“The sum A+B of two matrices A=[a,] and B=[b,] having the same order is the matrix obtained by adding corresponding elements Of A and B That is,

A+B=(s,]+Ið,) Matrix addition is both associative and commutative Thus,

AS(BSC)=(A+B)+C and A+B=B+A

(See Problem 12.)

“The matrix subiracion A~ B is defined similarly: A and B must have the same order, and the subtractions must be performed on corresponding elements to yield the matrix [a,~ b,]- (See Problem 13)

SCALAR MULTIPLICATION AND MATRIX MULTIPLICATION

For any sala (ths ook, wy mumber ora unctn of the matt HA (or, equalenty, At) Shain by mlupiog cre amen! of Ay the sear & Tat‘

rs] ={ka,] (See Problem 1.3.)

Lá NI Hộ Bờ] Ma ender 7p and pc, respectively o that te unter of cots na adr cane elements are Ben Dy ofA ste ter of ros of B Then epee AB Sheed to the at Each ment of AB is a do products obtied ty min te transpose of he ith ow fA fed then ings Jt pode wh the cto of B (ee bine toes 12) Matit meliicaon sano and dau ve son and aac eer iti sot comma Tas,

A(BC)=(ABJC-A(B+C)=AB+AC but, in general, ABBA Also,

ROW.ECHELON FORM

[A zero row in a matrix isa row whose elements are all zero, and a nonzero row is one that contains atleast one nonzero clement A matrix 2¢ro matrix, denoted O, ft contais Only 280 'A matric isin rom-ecelon form if it satisies four conditions:

(RI): All nonzero rows precede (thats, appear above) zero rows when bath types are contained in the matin

(R2}: The first (leftmost) nonzero element of each nonzero row is unity

(R3); When the first nonzero element of a row appears in column ¢, then all elements in column ¢

in succeeding rows are 260

(RU): The fist nonzero element of any nonzero row appeats in Inter column (further tothe right) than the ft nonzero clement of any preceding row

cur 1} BASIC OPERATIONS 3

Example 13 The mate

0323

00162

00000

‘ates al four consitions and 10 i in rowechelon form (See Problems 111 101.1 and 1.18)

ELEMENTARY ROW AND COLUMN OPERATIONS,

“There are three elementary row operations which may be used to transform matrix into row-echelon form The origins of these operations are discussed in Chapter 2: the operations themuelves are!

(Et): overchange any tvo rows

(EĐy Moliply the clemenlS of any r0 by 2 nonzero scalar

(£3); Add to any row, element by clement, 2 scalar times the corresponding elements of another

“Three elementary column operations are defined analogously

‘An algorithm for using elementary fw operations 10 transform a mati into row-echelon form is

18 flows:

STEP 1.1: Let R denote the work row, aed initialize R= (60 that the top row isthe fst work

te)

STEP 1.2: Find the fist column containing a nonzero clement i ether row R of any succeeding row If no such columa exists, stop; the transformation is complete Otherwise, let C

‘denote this column, STEP 1.3: Begioning with cow R and continuing through successive rows, locate the fist row

Inning 2 nonzero clement in columa C- If tis row is not row R, interchange it with row

2 (Clementary row operation EI) Row R wil now have a nonzero element in column

C This clement is called the pivot; let P denote its value STEP 1.4: IC Ps not 1, multiply the clements of row R by 1/ (elementary row operation E2);,

‘otherwite continue STEP 1.5: Seatch all rows following row R for one having a nonzero element in column C If 0

such row exists, go to Step 1.8; otherwise designate that row a8 row N and the value of

‘the nonzero element in tow N and columa C as V

STEP 1.6: Add to the element of row N the scalar ~V times the coresponding clements of row R

(lementary ow operation E3) STEP 1.7: Return to Step 15

STEP 1.8: Increase R by 1 If this new value of Ris larger than the number of rows in the matrix,

‘sop; the tansformation is compete Otherwise, return to Step 12

(See Problems 1.12 throngh 1.15)

RANK

“The rank (or row rank) of a mati isthe number of nonzero rows i the matrix afte it has been {ransformed to row-echelon form via elementary row operations (See Problems 1.16 and 1.17.)

of) HLg] eres

A-B=36) +6 +47)<0,

: e|j

Fat 94-058 he mae Pen 12

seas? Saft 31-[88 BI-ES 2S B38) 055)

[03 2 )-(5 35]

Find AB and BA for the matrices of Problem 1.2

-[# 1l & -7|*|ä&)+3@) 26)+XC|^las =n sH188:88 96) + I7} cle +

4 S|[0 1]_[ 40+52 - 40~568) ]„[ 10

aa-[s STL 3]*[«*C5e «0C 2@|*[-k -

Find AB and BA for

-Ít 23 "" 8)

Arld da] mm 9l s3

SSnce A has thee cums while B bạ on tro rons, the mate product AB i no defined But

[2 SIL } 3I»(sÃP†S 3`? †552» allt 011) 9K) 2) #98908) s5 2 -9N)

“ mo sớm

CHAP | BASIC OPERATIONS s

'A matrix is parttioned if itis divided into smaller matrices by horizontal or vertical lines

‘ann between entire rows and columns Determine thee paritionings of the matix

1203

78-1

“Ther te 2"— 1=31 diferent may in which A canbe partioned with teat one prtoning ine

By pacing» line between each io rows an each two columas, we vide Alto twee 1 1 maces, ching

By pacing one ine Deten th stand second rows and another ine Between the Second snd thi

‘ohaman, we conto the partioning

«hi

“hai cpa e-[8 9] Lt]

A thir parationing canbe constructed by placing 2 sings in ben tộc hid and fourth clumas ofA Then AG] where

BASIC OPERATIONS (eHAr

G-]00 5} T8 aa m=] 6 = [A paritoned matrix canbe viewed a 2 matric whose cements are themselves matrices

“The arithmetic operations defined above for matrices having salar elements apply a well 10 partitioned matrices Determine AB and AB if

where Ris anew row consisting ofthe column sums ofA, and Cis «new column consisting of

‘the row sums of B The resulting matrix has the original product AB in the upper lft partition Ifno errors have been made, AC consists ofthe row sums of AB; RB consists of the Column sums of AB: and RC is the sum of the elements of AC as well as the sum of the elements of RB Use this procedure to obtain the product

E 2l 2;

CHAP 1) BASIC OPERATIONS 7

`We on the pinidoned matces ad in thei produ:

toy an ary ol i fie 2 29)

112 Use elementary row operations to wansform matrices B, C, and of

Problem 1.11 into rowechelon form

We foto ig 1 rath Ln each cate at simpy lta ti ru mau anpin FB ch R= (ep 3) Sad CI ep TD oe agp ep 1 ad ————

ay (oi a]

sich is in romeeelon form, For matrix C, with R= (Step 1.1) C=2 (Step 1.2), and P=2 (Step 3) we apply Step 14nd mally all elements in theft row by 1/2, ohldng

Sa d|

sich ein rom-echelon form, For matt E, xi =1 (Step 1.1)

(Step 15) we appl Sep 11 by aang 0 each element in Yow 2,4 Ues the corespondingceme

ich ia rowel oom

1.13 Transform the following matrix into row-echelon form:

® [BASIC OPERATIONS [omar 1

38 9 0 2-102-2 Here (and in tter problems) we shall wie an stow fo indicate the rom that celts fom each

— fom operation

Đ 2 T3 -RỈ ayes Ads} tines he

k 2-1 i Step 16 with Ro 1, Cm 1, ¥=2, 2-1 2-2] row tome second row

212 8] Sa V=2 Aga ~P times the fet

[ 2-1 4] ‘Step 16 with R= 1, C=1,N=3,

Step 14 with R=2, C=2, and

lily the second ro

tyua Step 16 with R=2, Sed V= "5" Add Stumes the €-2, N=, road row othe tid rom

PL uz 052) step 1 with R= 1, C= 1, and

so tw 12

1120 siz slo 92 1 na S718

0 1 2/9 13/9] nd V=972" Aaa 972 times the

60 072 | Second som to the third ro

112 0 SIZ) step 14 with R=3, C4, and

01 2/9 13/9] P22 Matiply the tid roe by aloo 0 ca |

CHAP 1} BASIC OPERATIONS °

130 1/9 4/5 UV] Step 4: Mipty dhe Bet com

29 (oP Atl jot a7 byt

123 8 4/3 3] Step Lee Ad —2 times heft

0 53 -3A S19 1 BA S13] tow tothe second vow

23 US 43/3] Step LAL —1 times he Bt

5 “2's S13 “sis row wthe Mid vm

6 ĐA HH -mA mô,

130 TA -43 VI) Sep 16 Add 201 ies he #70) km aloo Wo

LÁT Determine the rank of the matrix of Problem 11S

Becawe the row-chelon form of his matrix hat tmo onsero ms, the ak of the eign matin a2

0 [BASIC OPERATIONS [cart

1.26 Find (o) BF and (6)

1.27 Tramforn A to rom-eclon form,

1.28 Teamform Ww rowechelon form

1.29 Tramforn € wo rom-eclon form,

1.0 Tranorm D wo row-estlon form,

1.31 Tramforn E 9 romechelon form

1.32 Fin the rank of (a) A: (6) B; (Q C; (4) D; s8 (9) E

zero mutre, whose product sa zero mates 1M The price schedule for a New York to Miami igh i given by the vector P= [240,10 $5], where the clement denote the cot of fst clas, business iss, and wurst clas cts, respecte The number

of ekts of ach clas purchased fora partir fight ven by the vector N= [8,21 13) Whats the Signicance of PNT

1.35 The iavemory of computers at each outlet of 2 threestore chain given by the mats

“1

\nbee the rows petin tthe ferent ores ad the columns denote the numberof brand X ad brand

Y computer espectively, in each soe The whoisle cont of thee computers ie en bythe ¥etor 1D [70 LAN)” Cakealate ND and sate gence

Chapter 2

Simultaneous Linear Equations

CONSISTENCY

‘A sjstem of simultaneous linear equations is a set of equations ofthe form

fut tah tenn _ + asp +

sm; j=1.2,-.-+m) and the quantities b, (=

Known constants The 3, ( {inate the unknowns whone values ace sought

"A solution for systom (2.1) isa set of values, one for each unknown, that, when substituted in the system, renders all ts equations valid (See Problem 2.1.) A system of simultaneous linear

‘equations may possess no solutions, exactly one solution, or more than one solo

‘The cocticients ay (

Example 21 The anton

has o solution, tecase there are no values for x, and, hat sum 101 and 0 simultaneous The system

bas the single alton x, 0,4, = 15 and

3ạ~z,=0 Ina sation, x," for every al of x

A set of simultancous equations is consisen if possesses at least one solution; otherwise iis

“The matrix A is called the coefficient marx, because it contains the coefficients of the unknowns

“The ith row of A (i im) corresponds to the ith equation in sytem (2.1), while the jth column “The augmented maivix corresponding of A (j= 1,2, ->n) contains all the coefficients of, one coefficient for each equation to system (27) is the partitioned matrix [A|B] (See Problems 22 through 2.4)

“ SIMULTANEOUS LINEAR EQUATIONS (CHAP 2

‘THEORY OF SOLUTIONS

‘Theorem 2.1: ‘The system AX = Bis consistent if and only if the rank of A equals the rank of [A |B]

‘Theorem 2.2: Denote the rank of A ask, and the number of unknowns a6, Ifthe system AX

consistent, then the solution contains nk arbitrary sealar

(See Problems 2.510 27)

‘System (2.1) ie said t0 be homogeneous if B =8; thai if By =B, =*+-=b, =0.UBA0 fie

it ar least one 8, ((=1.2, m) 18 not 2er], the system 5 nonhomogeneous Homogeneous

‘systems ate consistent and admit the solution x, x, - =x, 0, which i called the val solwiom'

1 nontrivial solution 1s One that contains at least one nonzero value

‘Theorem 2 Denote the rank of A as 4, and the number of unknowns as n The homogeneous

system AX =0 has a nontrivial soltion if and only ifm = k (See Problem 27)

SIMPLIFYING OPERATIONS

Three operations that

its solation Set are: ter the form of a system of simultaneous linea equations but do not alter (Oli: Ioterchanging the sequence of two equations

(O3): Malipying an equation by a nonzero salar

(03: Adding to one equation a scalar times another equation

Applying operations O1, 02 and O3 to system (2.1) is equivalent to applying the elementary

‘ow operations E1, E2, and E3 (see Chapter 1) tothe augmented matrix assoiated with that system Gaussian elimination isan algoithm for applying these operations systematically, to obtain a set of,

‘equations that is exny 10 analyze for consistency and easy to solve If tf consistent

GAUSSIAN ELIMINATION ALGORITHM

STEP 2.1: Form the augmented matix [AB associated with the given system of equations STEP 22 Use clementary row operations to transform {A |B] into row-echelon form (see Chapter

1), Denote the result (C|D}

STEP 23 Determine the ranks of C and [C| ] If these ranks are equal, comtinue the sytem is

consistent (by Theorem 21) If ot, stop: the orignal sjstem has no solution

STEP 24: Consider the system of equations corresponding to (C|D], discarding any identically

2ze10 equations (Ifthe rank of C isk and the numberof unknowns ism there will be

‘nk such equations) Sole each equation fr its first (lowest indexed) variable having

1 nonzero coefficient STEP 25: Any variable not appearing on the left side of any equation is arbitrary All other

variables can be determined uniguely in terms of the arbitrary variables by back

— (See Probiems 2.5 through 2.8 ) Other solution procedures are discussed in Chapters 3 4, 5, and 21

PIVOTING STRATEGIES

Errors due to rounding can become a problem in Gaussian elimination, To minimize the effet of roundoff errors a variety of pivoting strategies have been proposed, each modifying Step 1.3 ofthe algorithm given in Chapter I Pvoting srategics ate merely criteria for choosing the pivot clement

CHAP 3] SIMULTANEOUS LINEAR EQUATIONS B Partial pivoting involves searching the work colum of the augmented matrix forthe largest clement in absolute value appearing inthe current work row or a succeeding row That element becomes the new pivot To use partial pivoting replace Step 13 ofthe algorithm for wansforming a matrix to row-echelon form with the following:

STEP 1.3 Beginning with row R and continuing through successive rows, locate the largest

‘clement in absolute value appearing in work column C Denote the Bes tow in which Thức element appears as row J If i sillerent from R, interchange rows and R (elementary ow operation El) Row R will now have, in column C the largest

‘onzero element in absolute value appearing in column C of row Rot any row Succeeding it This clement in row R and column Cis called the pivot et P denote it value

(See Problems 29 and 2.10)

‘Two other pivoting strategies are described in Problems 2.11 and 2.12: they are successively more powerful but require addtional computations Since the goa! i to avoid significant roundof! ror, its not necessary to find the best pivot element st each Stage, but rather to avoid bad ones

‘Thus, partial pivoting ithe strategy most often implemented

3x, =1, and x,

.—

Br, xến văy=I Xin săn Substituting the proposed vales fr the woke ino the le side of each equation gies

2ayeu XS) 60) + CI0=1 - 'X8)©70) + IC10)=6

“The last equation does ot yield 8 a requred: hence the propoed values donot consti 2 slution

2.2 Write the sytem of equations given in Problem 2.1 asa matrix equation, and then determine its associated augmented matrix

SIMULTANEOUS LINEAR EQUATIONS

[nar 2 23° Write the following system of equations in matrix form, and then determine its augmented mates

3H t3 ty ,= 1 2m tầm aa ad xiên tần BS T

‘This system i equivalent tothe matrix equation

25 SoNe the set of equations given in Problem 2.1 by Gaussian elimination

“The augmented mati for this tem was determined in Problem 2210 be

21 ors lAlBi=[s 611 S718

kim.|t

18 fotows from Problem 1

sand har rank 2 Since the rank of C does st equal the rank of [C|D}, the orginal et of equstions

` "

fr, 0, e0, =1 and wich clearly has no solution,

CHAP 3] SIMULTANEOUS LINEAR EQUATIONS "

26 Solve the set of equations given in Problem 2.3 by Gaussian elimination

‘The augmented mattis for this system was determined in Problem 2310 be

3 caja]?

1

‘Using the reas of Problem 1.15, we transform this matrix nt the owecelon form

138 19 =43103) KelpiÌo + =5 +

ith xy and, abit

2.7 Solve the following set of homogeneous equations by Gaussian elimination:

o 0 lol lai

6

28

29

“The ra sĩ te cffdent matrix Ais thus 2, nd Beate there ae three unknowns in the rigia ct

‘St eqastons, the system his aon! solution The set of equations sociated with the supmented

“The augmented mutex sociated with thi system i

CHAP 2) SIMULTANEOUS LINEAR EQUATIONS ”

We write the system in matrix form, rounding 1 0001 to 1.000 Then we transform the augmented

‘maton ito row-echelon form ug the alot af Chapter 1, sn the folowing eps

(ote that we round t9 ~100000 twice ia the mex-o-ast step} The resuing augmented

‘mat shows that the Stem is consistent The equations asocated wit ts mutt ate

es aaltion i r=, 1, Wich ao the sottion to the orginal set of equations

‘Al computrs rund to a numberof wignicant furs & tat depends onthe machine being wed

“Then an equation of he form

TH ng th

2.10 Solve the following st of equations using partial pivoting:

x,42nj4 352 18

18 SIMULTANEOUS LINEAR EQUATIONS (CHAP2

Tetrseforningthế mai we peed owe Step 13 inmesiaely with R= 1 and C= 1 The ares,

‘lement in absolute vale im col 1s ~5, appearing in Tow 3 We interchange the rs and thd rows, {nd then continue the tenformaion to row ection form:

2.11 To use sealed pivoting, we fist define, asthe scale factor for each row of the coefficient matrix [Avthe largest element in absolute valve appearing in that row The sae factors ae computed

‘once and only once and, for easy reference, are added onto the augmented matrix [AB] a8 nother paritioned column Then Step 1.3 of Chapter 1 is replaced withthe following [Divide the absolute valve of each nonzero clement that iin the work column and of o below the work row by the scale factor for row The element Yielding the largest quotient the new pivot; denote its row as row I If row 1's different from the curreat work row (COW FR), then interchange rows I and R Row iterchanges ae the only elementary row operations tha are performed onthe scale factors: all ther stepe in the Gaussian elimination are himited

oA snd B

Solve Problem 2.10 using scaled pivoting

The sale factors fr the system of Problem 2.10 are

1,<ma(1.23)<3

=mst21|-4) =4

—

CHAP 3] SIMULTANEOUS LINEAR EQUATIONS ”

We add 2 column consiting of thee scale factors tothe augmented mati forthe system, and then transforming 0 Foweclon form 2 floes:

‘lementsn column I ate V/3=

Now ook fow is 2 and the work

‘column TS/3= 00 and 105/17 2 The qualens = O.8 ae

“The largest quotient 618 50 the pat 1.5, which appears fa tow 3 The second and ted tom

‘columns with unknowns must be implemented To do 0, add a new patttoned row, row 0, above the usual agomented ‘denote the subscripts onthe unknowas, will designate which unknown matrix Its elements, which are initially in the order 1,2 ` mt is associated with each

” SIMULTANEOUS LINEAR EQUATIONS jonar 2

‘ote the sytem of Problem 2.10 using complete pivoting We the bookkeeping rom 0 1 the augmented matrix of Problem 2:10, Then, Bpining wih ow I, we transform the remaining YOWS into ow-echelon form

Ra 1 and C= 1, The largest

‘ements abit value im the

OSes “TRIS | LUYNG

¬ work om and work column SiS] ate now R=? and Co? Tre -257163 | lagen element im able vale

‘Lesai2] ofthe fur under comideraon 28805, for whch =? and J = 2 i

Since [= and) = C00 Iimerchange vegies

"heft can of the resting rowechelon mat corresponds 9 £, and he thư cam lo xịn

‘the ascated set of equations

`

x,+0289lá., = =2 5718)

xi= LAN Ssving cach equation forthe fiat variable with 4 nonzero cefcien, we obtain

17 $6106 Oras, + 0.204,

25718 -0285745,

1 sor sich, when solved by hack substiution yes the solution, 1001, 5, =

“CHAP 3] SIMULTANEOUS LINEAR EQUATIONS a

1213 Gauss-Jordan elimination adds 3 step between Steps 2.3 and 2.4 of the algorithm for Gaussian climination, Once the augmented matrix has been reduced 10 row-echelon form i is then fedeced sill further Beginning with the last pivot clement and continuing, sequentially

‘backward tothe fst, each pivot element is used to transform all other elements in its column

‘Use Gauss-Jordan elimination to solve Problem 2.8

‘The teo ep: í (he Oaueian elimination algorithm ae wed to reluce the apmented mati

to rom-ehelon form ain Problems 113 and 2.8 Py

‘The st of equation associated wih this augmented matt ex, = 15,

solution set forthe opal sytem (no back subsutons equred)

° '

3.14 Use Gauss-Jordan elimination to solve the system of Problem 27

“The fast wo tps ofthe Gaussian cheiestion sp provide the augmented row-ecelon- oem

1a une Kia-|s TỶ “g503| ao oe

‘5m Problem 8 the (1.2) position: 2.7 Ths matrix seduce frter by sing the pivot inthe (2.2) position to ghế xe

2 SIMULTANEOUS LINEAR EQUATIONS tonar 2

Supplementary Problems LAS Which of

nensl ) x,=&x,<

7 22a are kvdom to the system

tint nes 3H m3 =I§

xi ST

2246 Wee the augmented mats forthe sytem gives in Problem 2.15

HT WaAe the augmented mate for ster,

ao

án cân 4, Spe 2

da, 8s, + ne, 1ar,= 10

1B Scie the set of equations anociated with each ofthe folowing augmented maties:

2219 Solve the system piven in Problem 2.15,

2220 Solve the system given in Problem 2.17

Problems 2.21 though 227, solve forthe unknowns i the given system

It Haye 6M quân vân thun R

4, *âm sân =S1 Dyn? nt ed

cue 2} SIMULTANEOUS LINEAR EQUATIONS 3

Use Gausian einai

then fd the ston to eteine values of & for which stuns exist othe following systems and

aut attra aan &

[A manufacturer produces three types of desks: custom, deluxe, and regular Each castom desk fequres 12 worker hous to ct and semble, and S worker Rou to khi Each dlone desk dregues

10 hours to cut and assemble, and 3 bouts to Reh each regular desk equies 8 hors to cut and

‘semble, and 1 hour to finish Ona daly basis, the manulactrer has avalble #40 worker hous for

‘Sung sod suembling, pd 120 worker hours for fang Show that he problem of determining box

‘many desks of each type to produce wo thu all workpowc is wedi equivalent 0 slvng two equations {he twee uoknowns c,d andr How many sctuons ae thee?

‘The eadofanesear employee bonus b 3 percent of taxable income f afer city and state tues are edocted The ely ti € i | peent of taxable income, while these tat # 8 4 petent of taxable Income with cet lowed forthe ey tax a2 pretax deduction Show that the problem of etecmining the bonus euialent fo solving three equations in the four unknowns B i, and

Prove that if ¥ and Z are two solutions ofthe Heat system AX.=B, then YZ is slution ofthe bomogeoeous system AX =

Prove that if ¥ and Z are eo solutions of the Bear sytem AX

soloon of the homogeneous system AX = 0, thee V2 M, where Mi

“The clements 44:5; dys 1dyy ie On ad fore the diagonal, aso called the main diagonal of

‘principal diagonal The elements 4, «immediately above the diagonal elemeats fore the superdiagonal and the elements "immediately below the diagonal elements constitute the subdiagonal

'A diagonal mais 4 square matrix in which all elemcats not on the main diagonal ae equal to ero; the diagonal elements may have aay values Aa idematy matric Ii 9 diagonal mate in hich

ll of the diagonal elements are equal 10 unity The 22 and 4» 4 identity matrices are

In mos cases, «square matrix A can be written asthe product of lower triangular matrix Land

an upper triangular matrix U, where L and U have the same order as A This factorization, when it feuist Is unique if the elements on the main diagonal of U are all Is That i

>

CHAP 3] SQUARE MATRICES 2

Crouts reduction is an algorithm for calculating the elements of Land U In this procedure, the fist column of Lis determined fs, then the frst row of U, the second column of L, the second row

‘of U, the third column of L, the third row of U, and so on uni all elements have been found The

‘order of Land U is the same as that of A, which we here assume iS n

STEP 3.1: Initialization: Ifa, =, stop: factorization is not posible Otherwise, the fist column

Cf Lis the first columa of A; remaining elements of the fst row of Lave zero The first

‘ow of Uis the fist row of A divided by =: temaining clement of the Sst column

‘FU are zero Set a counter at N=2

STEP 32: For i= N.N+1, m set L equal to that portion of the ith row of L that has

already been determined That is, L; consists of the fist N~ 1 elements ofthe ith row

kh STEP 3.3: For}=N,N+1, >m: set U; equal to that portion of the jth eoluma of U that has

already been deicrmined That is, U; consists of the rst NT elements of the jth column of U

STEP 34: Compute the Nth columa of L For each element ofthat column on or below the main

‘iagonal, compute

If any Iyy = 0 when NV % n, stop; the factorization is not posible Otherwise, set the remaining elements of the Nih row of L equal to 270

STEP 3.5: Set uyy = 1 IEN= n, stop the factorization isco

‘lements ofthe Nth column of U equal to zero and

‘element ofthat ro to the right of the main diag

(See Problems 3.4 through 3.6.)

Partial pivoting (see Chapter 2) is recommended when exact arithmetic isnot wsed and roundoff error i posible Prior to Steps 3.1 and 3.2 (for N= 2,3, -.), scan the Nth columa of A forthe largest element in absolute value appearing in that column and on or below the main diagonal I this clement is in row p, with p # N, then interchange the pth and Nth rows of A, as wel a8 the pth and [Mth rows L up to the Nth column (which represents the parts of those wo rows ia L that have already been determined)

SIMULTANEOUS LINEAR EQUATIONS

LLU decompositions are useful for solving systems of simultaneous linear equations when the number of unknowns is equal 19 the number of equations The matrix form of sich a system is

2% SOUARE MATRICES ICHAP.3

by reordering the equations, Gaussian clmination is applicable t all systems, and for that reason i often the preferzed algorithm,

Tf times the second row of A tothe third row of A

Since an elementary matrit is constructed by performing the deste elementary fom operation on

an ident matrix ofthe Appropriate size, th ate the 33 Henity, we Rave

CHAP 3] SQUARE MATRICES ” 33° Find a matrix P such that PA

STEP 3%

as

STEP 36: To this point we have

(-[L3} Ls) SFG srep 36

ca

CHAP 3} SQUARE MATRICES ”

36 Factor the following matrix into an upper triangular matrix and a lower tiangular matrix:

2210 30-11

¬

11 00, Using Crows reduction, we have

step 31:

2 3

tệ

“1 STEP 32 L;~[9),b5~[0), and

STEP 33 1.U,= [12], and; = 10)

STEP 34: fy = (LU; =0- [U1] = 0-0)

112 0 O1 86-10]

STEP 36: To this poit we have

Ee} ¬=~ 1 -sie 0] #H 8/00 1 cai 00 0<

Since N= and n= we increase N by 1 lo N S4,

S2hũng this system Sequential frm top 10 botiom we obtain y,

„ni ‘With these values and U a ven in Problem 36, xe can le (he system corespnding 5 y= 2609, y= 22/5, and to UX = ¥

tin Ine 2 tne # Sering i tem seuenty om etm 10, we bin estan whe egal sem

CHAP 3J SQUARE MATRICES a

‘38 Solve the system of equations given in Problem 3.7 i the right side of the second equation is changed from —1 te],

‘The coefficient matrix A is unchanged, 1 both Land U areas they were From (32)

39 Verity Crow's algorithm for 3% 3 matrices

Foc an 94a 3x 3 matic A, we Seek facoration ofthe form

ae en) Us fe tulle 0 ] Bt tín lạng hp Cụ =

By equating crtesponding coef io the order of fist column, remaining ft row, remaining

‘second colon emaining Second row nd emaiting third coma, and then Solving Soest tthe

‘ingle unknown i each equiion, we mou obai the formuls of the Crout redaction sigh,

2 SQUARE MATRICES [ears

f “8 4] Te i 3 ° 1 f ° 3 weosia-[ o's ‘o]-fo 3 |euja 4 6Ì-|s o6

3⁄12 A square matrix A is said to be nilpotent if A’ = 6 for some positive integer Test integer for which AP=0, then Ate sid to be nipXen o Indes pe Show that p If p is the least

i mipteat of index 3

1s 2

1241 36-3

“Tati indeed the ease, Deease

343 Fi elementary matrices that, when mliptied on the ight by any 3% 3 matrix A () wil interchange

te second and third ows of A () wl mp the fstrow ofA by 7; and (c) will a —3 times the fiat row oA othe second fom of JLU6 Fd elementary matrices that when multipied onthe right by ay 44 matic A (a) wil interchange Second and third rows ofA: (b) wil add 3 tenes the Hn ow of A 10 the out tow ofA and (2)

‘sit 2385 times the third row of & tothe Rist rom of A

[3.17 Prone thatthe product of two lower ings mates of the stme orders elf ower triangle

In Problems 318 through 3.23, wit each ofthe given matrices the producto lower triangular matic

sn an ope? tanga mate

“0 “H1 *I

CHAP 3] SQUARE MATRICES

Âm Tâm tâm tuy 16

(Hin: See Problem 3.19) 3 xâm cân xám, =8

(Hin: See Probie 3.21)

2330 Find A and A for the matic ghem in Probiem 215,

`

20 0) a-[or 0

oo

22 What doer A” look lie when A 2 agonal mater?

AY Asquire matrix Hi to be idempotent A= A Show that the folowing mati is Wempotet

“¡3

3.34 Prove that fA is Mlempotent, then 50 f00 I~ A

BAS Prove tat (A) =A)"

A square matrix i said tobe singular if it does not have an inverse; 2 matrix that has an inverse

is called nonsingular or inverable The inverse of A, when it exists, i denoted as A"

[SIMPLE INVERSES

Elementary matrices corresponding o elementary row operations (ee Chapter 3) ate invertible

‘An elementary matrix of the fist kind, one that coresponds to an interchange Of two rows, iis own inverse The imene ofan elementary mats of the ond Rnd, oe that responds miying

‘one tow of a matrix by a nonzero scalar k, i obtained simply by replacing the value of kia the tlementary matrix wih its reciprocal 1/k The inverse of an elementary matrix ofthe third kind,

self upper triangular, while that of a lower triangular matrix i lower triangular (see Problem 4.13), provided none ofthe diagonal elements is

2210 If at least one diagonal element is zero, then 00 inverse exists The inverses of triangular matrices are constructed iteratively, one column at atime, using Eq (4.1) (See Problems 4.3 and 44)

CALCULATING INVERSES

Taverses may be found through the use of elementary row operations (see Chapter 1) This procedure not ony yields the inverse when it exists, but also indicates when the inverse docs not Exist Am algorithm for finding the inverse of a matrix A is at follows

STEP 4.1: Form the partitioned matrix [AI] where His the ideatty matrix having the same order

BA

STEP 4.2: Using elementary row operations, transform A into row-echelon form (see Chapter 1),

applying each row operation tothe entire partitioned matrix formed in Step 1 Denote the result a8 [CD], where C is in row-echelon form

STEP 43: If Chas a zero row, top; the original matrix Ais singular and does not have an inverse

‘Otherwise continue the orignal mattx is invertible

*