LECTURE )4CONDJTJONJNG OF Tf-IE LEAST

sa UA1tEs PROBLEM/,

L ec/s ı. s'1uores r::c.». A E c'" (wilJ rn > (/)anel b 6. cem. Fi(JJ x 5fJch IJa.f

ii Ax - b LL ls as 5mcJJ as r=":

~~9Ax=.!i

raJ13e rA)r

RercJ1 thal Of fimcJ X 5all5fies

Ax b-AyIn o/her f))ord5 Ax is {he

orülO} ofld ?'oj ed/of) 0/ b onlo IOfIJe c )

.!J = A X (5ıuh that ıı. /;-3)f:hQ. od hogoiJa1 F/'tj eel/dil 0/ b (}il/o ratYrA)(j)

5e(}5iVii!} tUI?S-LOfJS

*' J--Iow do eS}j ==r*' /-Iow c/o es X chorıjt:k !-Lo!» dO€'5 3' chClIlJ e

*' How JoeJ )( cJU1}rz

bb?,

A7r

A7,

/))' r. {,

LA), r: Irw. r: irWr r; t.

N orrncJ. efu cıtJ0I15

The 5en5/V/ty orıcJ.Jse5 art! bas(J) 011

the f1Nmd erua D'OJ1

A~Ax =Akb-I-hal is sati5{led b; the j'ojlılro>1

)( Lo L5P~

P5e u clD j r1 ve r S e..

The 50JuLiM 10 L5P (arJ be tAJrıtter] tU

(o.5svrnit1j A hos {ıJ1 ronk)X = (dkAri A* b

--,---rıxoı (YI)( n (J x ın

1he =:": ver se Of A r= ([1TI)(fi

Af- = (A~AtA ~ e ([()(rY)

ExA MPLEl.e! J' :==::

o2.-

oif oo 4 /

Therefore. A1"=(A1<A)-A*= '3 O

O i~

rf< o fE/( TL f- 5-T *

'{l)w(A+r- = A @ (A~) =(A+) .(2) (Al3t = BtAi-(3) ı.e ': A6([wfl

wil-hmjnand ra (J k ( A) ::::::-n, 1h e J1

A+A = Tn501u1iofl 0/ LSP it1 İerms 1-r-::

)(= +-bj = A+b

P5eudo /l1vef5'e j!7 Larrıs o!:. 5VD

ı..,

be

A=O~v*- / ) )mrn I1xl1 nx n

reducec/ S /D o/

,D ~c7()

AE CrnXf1(with)tn ~{L

cJteri1C1iIVi? 0t1c1 etj/JivcJerdJUlt'/3df2I/r1~ -LAe r-r=: l' A 15

--1- /'\"A + A-*i- zzzz: V ~ U

Anto

o

/ / r7J O:J =F cO (77==0

J

.J 1?t1 e/al cieF;1/ bo iJ/"

tank ( ) -: ()({h iS 15 tA mor eil: hoJe/; mA en

1her~{o('er'''V tl c for

( i )(2)

AA+- iS {he (Jrth!YJ'CJt1~

ol1f" rat!je (A).

(AA+)* AA-f(A A+) (AAf)=(A (A*Ar'Aj(A(Aj<ArA~)

= A (A~ArAt:szzz: A A 'i-

EX AlvfPlE

Le L

A====3 DO 2-

D c

. -'~

AT~ljj' O O- () ~ 6 '

~

/5

o OO i OO O O

lht or!:hOj'O/lcJ /ltJNlorr(,lYij e (A) =- 5f°t1

}en5iVilj 0/ L5P w.r.L b5e IJ 5 i vi!:..!J cleteııels an //ıtee ri/IYf//; tle 5

( i) CtJlJJi/jo!l t1umber oL A {;CrtlXf)

K (A) = Il LL ii ~Jt= <:

~

(2) Afljl{' beilAJuiJ h tllld 3·b

(3) C}oser;es5cd0Yl3 x·

IlA iL 11)(IL!L- ıllıxiI 2...

REHARK- * /i A X LLL < iA iLLi X LLL 1 tl"> i1< fAx ıı = iiA 1/2. /ii i~ dllJy

L r « ooin 's in lh e ciire.L , j' D 17 ;?l-!(Ji r/~hı .JiIJ3u!/lr veel"r aSf(}(/alf) w/th OJ .. V;V

@ The sefI5iV/& 0/ J W· r. L b!j=AA+h

vieıo 3 as tl /Uf}C!: iM 1h

REHARk5(J) çar t/7 )i/lfar (LAflcbon

+- (x) = A X

-Uıe ~O[Dbtot1 matrix i5 {t?fIf-LClII'( re,~ (x) =A

e-- ~

~

.-. ~

*- ~

, ~ ~ ~

<:»

~ ~ ~ - ~

iv, >« '-"l -..

ı., ~ rJ)~ ~

~,

'""

~(~ ~

~ J:::>~

~ 4-

\

?:;( ~ ~<.ı,

\\>< < ~

>(

- ı

'-ı \i ~ ı\r\) ~

+

- .~

j-J

- 11--)'-l

- ~-4-

~ ~ ~~

~)ez A

- -+

"'5

LL

-- ~.

- +-x -- tr-

~ X

- v...

- - <c:»

\..,{

~ ep=-- ':;;:::t'J

f)

- + - ~

f'"ı-...... X

- es--:)

"-..-

\1-

r

- rı-t::> ~ r-

~ - ~

.. ~

~)

~

- --- ~-- ~~

t'i-- rI'\ ~" ~

, x+'"'>

~ x~\ı ~

~\-~

<:::::.

'x~

............... ---~ ~

~~

~

~x:

:?Q-=-"----'

f{ = fe ( ) ( ) (II x i )(05 CT Al IX

~K)(05 (!)- , n.

5ummar!JSen; iv" '-3 ez{- X ıo. r. t . b

iI = f{(A) !t(' ;rı )(05 LT' IL

CO/\/CLU510/\15

The so}uilMI o/ LSP is m»reS ens i 1. ive -lo teY t urh4 t/O/?3 LLL b whı?r1

* }-C{A) 15 JM'je; -LJı~ matri),'

A is ;11- can dı'/Jo/?(Jdj

ok Co5 (J- is {}Off' to 2NO; Lh e.

vee {Clr b ;5 c'joJe to be //1yrerte/?)(cuJor Lo raJ1.f{Z (A).

5eJ15 rtı -iy 0l LSP IA/J: t. AThe se!ıs;(;vi& olıcJ.J.fe5 Ort' rnor e//Jl/(}')vecl wr.I:,A.We LU/)) sh~ IAe.

JerivafroM 0/ il.

~

/{3 tA/F rr t. A_

«:A)/coJ (f))

X uırr{r A2.

f{(A) lon% +/({A)

5ee taJflS /33 ~/3-:'fOI clerlVt1{ofU). /il /reı~l-he» le Bcı 1-/

AJt1i/1 the J'tJluf)OI1 0/ LSP is 5MsibveLo ter ttJrhcrtfot?S 111 A WAM

7( L5 i11- co/l);-!JoJ1e~-% lı is L}oJe to bel'!}

f iJ-r;ıendiLular to rot?.1e (A).

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