Introduction. Farkas Lemma : Let A. System 1 : There exists x R. System 2 : There exists z R

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1 Intoduction Fakas Lemma is a classical esult, fist pulished in 9. It elongs to a class of statements called theoems of the altenative. hese theoems ae of the fom : one of the sstem and the sstem has a solution ut neve oth has solutions. his theoem chaacteies the optimalit conditions fo some optimiation polems. It is also used in poving the dualit theoem in linea pogamming. he othe theoems of altenatives ae Godan s, Gale s, Stiemke s, Motkin s, Da s, Ke s, etc. Fakas Lemma : Let R mn and R m. hen one and onl one of the following sstems has a solution : Sstem : hee eists R m, > such that. Sstem : hee eists R m, <, > he chapte contains peliminaies fom [6]. he well known weiestass theoem is poved fist and then the closest point theoem. Fakas Lemma is poved as a coolla to the fundamental sepaation theoem. Vaious definitions like hpeplane, sepaations, conve cones etc. ae given. Futhe, it is also poved that a conve set has a suppoting hpeplane at each ounda point. n optimal solution to a linea pogamming polem is a suppoting hpeplane and eve suppoting hpeplane contains an eteme point of the conve set of feasile solutions. Net, fo a non-empt closed conve cone we pove C C**. he Fakas Lemma is simpl eithe c C o c C. ()

2 he chapte is n pogamming polem, algeaic poof of Fakas Lemma []. Conside a linea Minimie c Suject to > > Its dual is Maimie Suject to < c > B weak dualit theoem fo feasile &, we have c > B stong dualit theoem if we could find and such that c then and ae optimal solutions. Hence, inview of weak dualit theoem it is sufficient to find & such that c <. his ineualit can e added in eisting constaints of a given linea pogamming polems and we can ignoe optimiation aspect. hus, if we can find a vecto u > such that Bu >, whee B c c, u t and t > ; afte scaling of u such that t, and ae optimal solutions of the pimal & the dual espectivel. Finding such a vecto is called ucke s theoem. he main theoem poved is : ()

3 Let Q e an othogonal mati. hen thee eist a vecto > and a uniue sign mati S such that Q S, euivalentl SQ has an eigen value. Note that the Cale tansfom states that if B is a skew smmetic then Q (I B) (I B) is othogonal. he ucke s theoem is poved using the main theoem and othe theoems of altenatives ae poved using ucke s. ()

4 CHER eliminaies We sa that is a minimiing solution fo the polem min {ƒ() : S}, if thee is S, ƒ( ) < ƒ () fo all S. Weiestass heoem : heoem. : Let S e a non empt, compact set and let ƒ : S E e continuous on S. hen, the polem min {ƒ() : S} attains its minimum, i.e. thee eists a minimiing solution to this polem. oof : Since ƒ is continuous on compact set S, ƒ (S) is compact and henceƒ is ounded elow on S. ( [8],4.5) Since S φ, then eists a geatest lowe ound, α inf {ƒ() : S}. Now, fo < <, conside the set S k { S : α ƒ() α k } fo each k,,.. definition of infimum, (α k in not a lowe ound), S k φ, fo each k. Choose k S k, k } k fo each k,, to get a seuence {. () 4

5 Since S is ounded, the seuence is ounded, ([8],.6), thee is a convegent suseuence { n k S. B continuit of ƒ, ƒ ( }, which conveges to sa. Hence is a limit point and S is closed, so ) ƒ ( ). Since α ƒ( n k n k ) α k k, we have α lim ƒ ( n k k ) ƒ ( ) hus, is a minimiing solution of the polem min {ƒ() S}. Closest oint heoem : heoem. : Let S e a non empt, closed conve set in R n and S. hen thee eists uniue point S with minimum distance fom. Futhe moe is the minimiing point if and onl if ( - ) ( - ) S. oof : Eistence of a closest point. Since S φ, thee eists a point eual to. Hence we conside the set, S. cleal minimum distance of S fom is less then o S S { : < } instead of S. Note that S is compact as it is closed & ounded. s a nom is a continuous function, Weiestass heoem, thee eist a minimiing point in S that is closest to the point. Uniueness : Suppose thee is an such S such that γ.. () () 5

6 B conveit of S, S. hen iangle ineualit, we get, γ If stict ineualit holds, we have a contadiction to eing the closest point to. heefoe, eualit holds and we must have, Now,( ) implies λ. If λ, then - λ ( - ), fo some λ ( - ) ( - ) S. Contadiction the assumption that S. Hence, λ i.e.,. Now, suppose that ( - ) ( - ), fo all S. Let S. hen ( - ) ( - ) - [aallelogam law] s - and ( - ) ( - ) his means is the minimiing point. Convesel, assume that is the minimiing point i.e., Let S and note that conveit of S, - - S. () 6

7 λ ( λ ) λ ( - ) S. fo λ. heefoe, - - λ ( - ) -.() lso, - - λ ( - ) () and () implies - λ - λ ( - ) ( - ).() (aallelogam Law) - λ - λ ( - ) ( - ) - λ ( - ) ( - ) λ - (fo all λ ).(4) Dividing e n (4) an such λ > and letting λ, we get ( - ) ( - ). Definition. : hpeplane H in R n is a collection of points of the fom, { : p α }, whee p is a noneo vecto in R n and α is a scala. Note : Since fo, H, p ( ), the vecto p is called the nomal vecto of the hpeplane. Remak : hpeplane H defines two closed half spaces, H { : p α } & H { : p α } & the two open half spaces, { : p > α } & { : p < α } () 7

8 Definitions.4 : Let S and S e non empt sets in R n. (a) hpeplane H { : p α } is said to sepaate S & S if p α fo each S and p α fo each S. () If S S H, then H is said to popel sepaate S and S. (c) he hpeplane H is said to stictl sepaate S and S if p > α fo each S and p < α fo each S. (d) he hpeplane H is said to stongl sepaate S & S if p > α fo each S and p < α fo each S, whee is a positive scala. Fundamental Sepaation heoem: heoem.5 : Let S e a non empt closed conve set in R n & S. hen, thee eists a noneo vecto p & a scalaα such that p > α & p α fo each S. oof : he set S is a non empt closed conve set and S. Hence heoem., thee is a uniue minimiing point S such that ( - ) ( - ) fo each S. Let, p ( - ) and α ( - ) p. () 8

9 hen, ( - ) ( - ) ( - ) ( - ) ( - ) ( - ) ( - ) i.e. p α and p α ( - ) ( - ) ( - ) ( - ) - > p > α [ S] Coolla.6 : Let S e a closed conve set in R n hen, S is the intesection of all half spaces containing S. oof : Let H {H i S H i, i I} is a collection of half spaces.cleal, S H i i I Suppose H i i I S. Let such that S. B fundamental sepaation theoem H i i I thee is a vecto p & a scala α such that p > α & p α fo all S. Denote Hα the half space p α. his half space must e in the collection H. Hence Hα, which in not tue. () 9

10 Coolla.7 : Let S e nonempt set and let hull. hen thee eist a stongl sepaating hpeplane fo S and. oof : ake S cl (conv S) cl (conv S), the closue of the conve heoem.8 (Fakas Lemma) : Let e an m n mati & c e an n-vecto. hen, eactl one of the following sstems has a solution. Sstem : and c > fo some R n Sstem : c and fo some R m oof : Suppose that sstem has a solution. i.e., thee is > such that c. Let e such that <. hen c [ c & ] hus thee is no s.t. < and c > i.e., sstem has no solution. Now, suppose that sstem has no solution.let S { :, } Cleal S is a conve set. B definition of S, it is an finite inte section of hpeplanes a j j, whee a j is the column of and a half space >. ll ae closed and hence S is closed. p We have c S, fundamental sepaation theoem, thee eists a vecto n R and a scala α such that p c > α and p < α fo all S and so p c >. lso, fo all >, α > p p. Since > can e made aitail lage, p <.. Since S, α > hus, thee is a vecto p R n such that p < and c p > i.e. sstem has a solution. ()

11 Geometical Significance of Fakas Lemma : We know that one & onl one of the following two sstems has a solution. Sstem : > and c < ; Sstem : w c and w > ; Whee is a given m n mati and c sstem can e witten as n R.Denote the i th ow of a i, i,... m. he m w c, w i > fo i,.. m. i i i a In othe wods, Sstem has a solution if and onl if c C, the cone geneated the ows of. Fakas Lemma assets that eithe c C o c C. In the fome case, sstem has no solution, and in the latte case, it does. Euivalentl, fo, we get sstem : <, c >. Geometicall, such a vecto should make an angle > 9 with each ow of, since a i < fo i,.. m, and it should make an angle < 9 with the vecto c since c >. heefoe, sstem has a solution if and onl if { : < } { : c > } φ. Figue (a) shows sstem has a solution. (with an in the shaded egion ielding a solution to sstem ). Figue () shows sstem has a solution. ccodingl, sstem has no solution in figue (a) and sstem has no solution in figue (). ()

12 Coolla.9(Godan s heoem) : Let e an m n mati. hen eactl one of the following two sstems has a solution. Sstem : < fo some R n ; Sstem :, fo some R m ; oof : Note that Sstem can e euivalentl witten as es fo some R n whee s >, s R and e is a vecto of m ones. Rewiting we get, [ e ] s and (,,) > s fo some R n. Now, the associated Sstem in Fakas Lemma is, s e (,,) and fo some R m.i.e.,, e and fo some R m. ()

13 his is euivalent to Sstem of the Coolla. Hence the esults follows Fakas Lemma. Coolla.: Let e an m n mati and c e an n vecto. hen eactl one of the following two sstems has a solution. Sstem : Sstem :,, c > fo some R n c, fo some R m oof : Wite Sstem as I c, [ ] he associated Sstem in the Fakas Lemma is, c > - I Fo some R n ; which is in fact,, c > the Sstem in the Coolla. Hence esult follows Fakas Lemma. Coolla. : Let e an m n mati, B e an l n mati and c e an n vecto. hen eactl one of the following two sstems has a Sol n. Sstem : Sstem :, B, c >, fo some R n B c,, fo some R n and R l oof : We can wite Sstem as B - B, c > he associated sstem in Fakas Lemma is ()

14 [, B, B ] c, i.e., B B c. Which gives, B c,, his is sstem of the coolla. Hence esult follow Fakas Lemma. Suppot of sets at ounda points : Definition. : Let S e a nonempt set in R n, and let S(the ounda of S). hpeplane H { : p ( ) } is called a suppoting hpeplane of S at, if S H, i.e. p ( ) fo each S o if S H such that p ( ) fo each S. heoem. : Let S e a non empt conve set in R n, and let S. hen thee eists a hpeplane that suppots S at ; i.e., thee eists a noneo vecto p such that p ( ) fo each cl (S). oof : Since S, thee eists a seuence { k }, k cl (S) such that k.b fundamental sepaation theoem, coesponding to each k thee is a p k with p k such that pk k > pk fo each cl (S).[In fundamental sepaation theoem, the nomal vecto can e nomalied dividing it its nom so that p k ] Since {p k } is ounded, it has a convegent suseuence { p is also eual to. Consideing this suseuence, we have (S). hen lim ( k p n k n k p ) > n k n k } with limit p whose nom p n n > p fo each cl k k nk p ( ) > ( n k ) 4 ()

15 i.e. p ( ) his is tue fo each cl (S), then p ( ) is a euied hpeplane. Coolla.4 : Let S e a non empt conve set in R n & int (S). hen thee is a noneo vecto p such that p ( ) fo each cl (S). oof : Note that R n et (S) int(s) S. If S then cl (S) int (S) S. he coolla follows fom the fundamental sepaation theoem. If S, then coolla follows fom the aove theoem. Coolla.5 : Let S e a nonempt set in R n, and let int( conv (S)). hen thee eists a hpeplane that sepaates S and. oof : ake S as conv (S) in aove coolla. Coolla.6 : Let S e a nonempt set in R n, and let S (conv(s)). hen thee eists a hpeplane that sepaates S at. oof : ake S as conv (S) in aove coolla. he Sepaation of wo Conve sets : heoem.7 : Let S and S e a nonempt set in R n, and suppose that S S is empt. hen thee eists a hpeplane that sepaates S and S i.e., thee eists a noneo vecto p in R n such that. inf {p : S } Sup {p : S } oof : Let S S Θ S { : S and S }. Note that S is conve, as S, S ae conve sets. Futhe moe S ecause othewise S S will e nonempt. B coolla.4, thee eists a noneo p R n such that p, fo all S, i.e. p p fo all S, S inf{p : S } p 5 ()

16 inf{p : S } Sup {p : S } Coolla.8 : Let S and S e nonempt conve set in R n. Suppose that int(s ) is not empt and that S int (S ) is empt. hen thee eists a hpeplane that sepaates S and S ; that is, thee eists a noneo p such that inf {p : S } Sup {p : S } oof : In the aove theoem, eplace S int(s ), then inf {p : S } Sup {p : int (S )} Sup {p S } Coolla.9 : Let S and S e nonempt sets in R n such that int (conv(s i )) φ, fo i,, ut int (conv(s )) int (conv(s )) φ. hen thee eists a hpeplane that sepaates S and S. heoem. (Godan s heoem): Let e an m n mati. hen, eactl one of the following sstems has a solution. Sstem : <, fo some R n Sstem : p and p fo some non eo p R m oof : We shall fist pove that if sstem has a solution to sstem.suppose on the conta that a solution some noneo p R m., then we cannot have a solution p eists i.e. p & p fo <, p and p p <, p < 6 ()

17 But this contadicts the hpothesis that p. Hence, sstem cannot have a solution. Now assume that the sstem has no solution. Conside, the following two sets, S { :, R n }, S { : < } S and S ae nonempt conve sets. Cleal if S S φ sstem has a solution. hen theoem.7 (sepaation of two conve sets), thee eists a hpeplane that sepaates S & S i.e., thee eists a noneo vecto p such that p p, fo each R n and cl (S ). p ( ) Since each component of can e made an aita lage negative nume, we must have p. aking, we have p, fo each R n. B choosing p, p ( p) p Hence sstem has a solution. p heoem. (Stong Sepaation) : Let S and S e closed conve sets and suppose that S is ounded. If S S is empt then thee eists a hpeplane that stongl sepaates S and S i.e., thee eists a noneo p & > such that inf {p : S } Sup {p : S } oof : Let S S Θ S and note that S is conve and that S. Let { k } e a seuence in S such that k. B definition of S, k k k, whee k S and k S. Since S is compact (closed and ounded), thee is a suseuence { Weiestass popet). hus, with S and S ; so that S; and S is closed. k n } with limit in S (Bolano k n. Since S is closed, S. heefoe 7 ()

18 Now, fundamental sepaation theoem, thee is a noneo p and an such that p fo each S and p <. heefoe >. B the definition of S, we conclude that p p fo each S and S. Coolla. : Let S and S e non empt sets in R n and suppose that S is ounded. If cl(conv S ) cl (conv S ) φ then thee eists a hpeplane that stongl sepaates S and S. Conve Cones and olait : Definition. : non empt set C in R n is called a cone with vete eo if C then λ C fo all λ. Definition.4 : Let S e a non empt set in R n. hen the ola cone of S, denoted S*, is given {p : p fo all S}. Lemma.5 : Let S, S and S e non empt sets in R n. hen the following statements holds tue.. S* is a closed conve cone.. S S**, whee S** is the ola cone of S*.. S S, implies that S * S *. oof :. We know, S* {p : p, fo all S} Let, S*. Fo λ ( - λ ), p p (λ ( - λ ) ) S* λ p ( - λ ) p 8 ()

19 Net, fo λ p λ p p (λ ) λ S* Hence, S* is a cone. Futhe let e a limit point of S*. hen thee eists { k } S* such that k. We have p k. Since inne poduct is continuous p k p. If p > then p fo, thee must e some p k (p, p ). Since p p k >, a contadiction. Hence, S* is closed conve cone. p >,. We have, S** { : S*}. Let S. Fo all p S*, p i.e., S**.. Suppose S S. Let p S * {p : p, S } p S p S [ S S ] p S * heoem.6 : Let C e a non empt closed conve cone. hen C C**. oof : Cleal C C**. Now let C** and suppose contadiction that C. B fundamental sepaation theoem, thee eists a noneo vecto p and a scala α such that p α, fo all C and p > α. But since C,α and so p >. If p C*, then p >, fo some C and p (λ ) can e made aitail lage choosing λ aita lage. his contadicts the fact that p α, C. heefoe, p C*, since C** {u: u v, v C*}, p, which is not tue. Hence C. 9 ()

20 Fakas heoem as a conseuence of aove theoem : Let e an m n mati, and let C { : }. Note that C is a closed conve cone, cleal, C* { : } c C** c fo all C* [ c ] Now, c C c,. hus, C C** can e stated euivalentl as follows : Sstem has a solution if and onl if Sstem has a solution. Sstem : implies c Sstem : c, he aove statement can e put in the moe usual and euivalent fom of Fakas theoem : Eactl one of the following two sstem has a solution. Sstem :, c > (i.e., c C** C) Sstem : c, (i.e., c C) ()

21 CHER n algeaic poof of Fakas lemma he Cale Connection: Let R mn, R m, and c R n, all e aita. hen the pimal L polem is : min c, > and >.(p) he dual is, ma, < c and >..(d) Weak dualit heoem : If is an feasile solution to (p) and is an feasile solution to (d) then < c. oof: If oth and ae feasile then < () c i.e., < c...() hus if feasile vectos * and * can e found that satisf c * * then * and * must e optimal solutions of oth the pimal and dual polems. ()

22 Stong dualit heoem : If * is a feasile solution to (p) and * is a feasile solution to (d) such that c * * then * is an optimal solution to pimal and * is an optimal solution to dual. he idea is to ignoe the optimiation aspects and add to the eisting constains the ineualit c <, ecause weak dualit theoem, we alwas have < c hus if we can find a vecto u > such that B u >, whee B -c - c -, u. t hen,we have Bu - ct - t - c Fo t >, u ma e scaled so that t. his gives >, > [ u > ] c >, > and > c hus, fo an skew smmetic mati B, if we can find a non negative vecto u such that Bu is also non negative then this vecto will give the solution to the geneal L polem.he eistence of such a vecto is guaanteed ucke s theoem. ucke s theoem : Let e an aita skew smmetic mati. hen thee eists u > such that u > and u u>. ()

23 he main theoem in the dissetation is aout a popet of an othogonal mati which is elated to the skew smmetic mati the Cale tansfom. he Cale ansfom : If B is skew smmetic then Q (I B) (I B) is othogonal. oof : Let C (I B) then Q C (I B) C CB C CB (B B ) (I CB C ) C...() Now, I CB C I (I B) B (I B) I (I B) (B B B) I (I B) (B BB ) (B B BB, since B is skew smmetic) I (I B) (I B)B I B I B hus, Q (I B) C (I B) (I B) (I B) (I B) Now, Q Q [(I B) (I B)] [(I B) (I B)] [(I - B) ((I B) ) ] [(I B) (I B)] (I B) (I B) (I B) (I B) (I B) (I B) (I B) (I B) ()

24 I Similal, Q Q I s eigen values of a skew smmetic mati ae puel imagina, (I B) B i µ ( i µ ), [ µ eal] heefoe, eigen values of I B have the fom ± i µ j, whee µ j is eal. Futhe, Π ( i µ j ), I B is alwas non singula. Now, Let λ e an eigen value of Q. Suppose B µ j. hen, Q (I B) (I B) (I B) ( B) (I B) ( i µ j ) ( B i µ j ) (I B) ( i µ j ) ( i µ j ) (I B) ( i µ j )( i µ j ) [ λ, & if is invetile, then λ. i.e., λ is eigen value of.] his shows that Q will have eigen values eual to ( i µ j ) / ( i µ j ) with eigen vectos as that of B. iµ Since iµ j j, the eigen values of Q lie on the unit cicle in the comple plane. Osevation : 4 ()

25 o eve skew smmetic mati B thee coesponds an othogonal mati Q (I B) (I B). But, thee ae matices Q that ae not Cale tansfoms of an skew smmetic mati. Conside a skew smmetic mati of ode, B -θ θ Whee θ is an aita eal scala. B Cale tansfomation, Q (I B) (I B) θ -θ -θ -θ -θ θ θ -θ - θ θ θ -θ Note that the diagonal elements of Q must have same sign egadless of the choice of θ. Hence no othogonal mati whose diagonal elements have diffeent signs can e epessed as a Cale tansfom. - Fo eample, if Q then thee is no skew smmetic mati B such that Q (I B) (I B). Definition. : sign mati is a diagonal mati whose diagonal elements ae ±. he Main heoem: heoem. : Let Q e an othogonal mati. hen thee eist a vecto > and a uniue sign mati S such that Q S. OR 5 ()

26 () 6 Fo an othogonal mati Q thee eist a uniue sign mati S such that SQ has an eigen value unit with a coesponding stictl positive eigen vecto. oof : he poof is induction an ode of an othogonal mati. If m, then fo Q [α], Q Q Q Q implies α ±. Hence take S Q fo an >. Suppose the theoem is tue fo othogonal matices of ode m. Let ( ) ( ) m m R Q e othogonal and let Q whee R m m. Since Q Q I, I m. We have () I m and I QQ m If, implies. Hence. Futhe similal gives. hus, in this case Q

27 () 7 and S S m whee S m is a sign mati of ode m which eists induction hpothesis. Hence assume that < and fom e n (). Let Q and Q. We claim that oth Q,Q ae othogonal. Fo eample Q Q... I ( ),, I ( ) ( ) ( ) ( ). I Similal, Q Q I. nd,

28 () 8 Q Q.. ( ) ( ) ( ). I ( ),, ( ) ( ) I () I s ode of Q, Q is ode of which is one less than ode of Q, the induction hpothesis thee eist positive vectos and and sign matics S and S such that Q S and Q S. So that, fom euation (), ( ) ( ) Q Q S S.() (S is sign mati, S S ) Now, thee ae two cases : S S : I Case.

29 ii ii ii ii o ( Sii and Sii ) hee eist i, S S i.e., ( S and S ). hen, in the poduct SS, the ent i i < and hence SS <. hus fom euation, ( )( ) - S S >. and, and oth must have the same sign. Define η, η Now, fo and, η η We have Q η η s, η Q S ( induction hpothesis) and, η η We have 9 ()

30 () η S Q. Similal η S Q. Now, since <, < and >. Hence, since and oth have the same sign, one of η and η is positive and thus one of and is euied vecto. Fo eample fo, η S Q Case II : S S I S S S S heefoe in euation () one of and is eo. We ma assume without loss of genealit that. So that. Q S and hence σ S S i.e., we ma wite, (a) S Q ˆ

31 () whee σ S Ŝ and ± σ is undetemined since the last elements of and Q ae oth eo. Now, epatition Q so that m m whee Q R We epeat the pevious agument. hus, in this case. Q, Q ae othogonal. Futhe, I Q Q Net, as aove in case of S S, induction is estalished. In case of S S ; again as aove thee eists a positive vecto and a sign mati S such that S Now, fo, () S Q ˆ Whee σ, S σ S ± ˆ dding the euation (a) and (),

32 Fo ( ) Sˆ Sˆ (4) Q j m, the j th elements of oth and ae stictl positive. So that if, fo an of these elements, the coesponding diagonal elements of Ŝ, and Ŝ ae diffeent, ( ) S Sˆ < Q ˆ, ecause, if i th diagonal elements of Ŝ and Ŝ ae diffeent then the i th tem in the epansion of S Sˆ ˆ will e ( ) while i th tem in the epansion is ( ) i i. i i Since othogonal tansfomation peseve lengths, we must have ( ) Q. heefoe, fo j m elements of Ŝ, and Ŝ ae same. s we ma choose σ, σ aita, we ma choose them so that Ŝ Ŝ. hen the euation (4) ecomes. ( ) S ( ) Q ˆ. Whee is stictl positive and Ŝ is a sign mati. his completes the case whee S S. Now, to show that S is uniue. ssume that thee eist two positive vectos and and sign mati S and S whee S S such that Q S and Q S. hen, ( S is sign mati. S ) Q Q SS S ut if S S we have, SS <, a contadiction. Coolla. : If is diagonall simila to some othogonal mati Q then thee eist > and sign mati S such that S. oof : Let D - QD whee D is diagonal. Let S e a sign mati such that D SD is non negative. ut Q SQS.Note that Q is othogonal. Hence the theoem., Q S fo some and S. ()

33 Fo D oseve that is positive. ( D QD) ( D ) ( S D ) ( S QS ) ( S D ) ( D ) ( D Q ) ( ) ( D S ) SD a [ D S S D ] S m n heoem.4 : Let V R whee m n and let V V I. hen thee eist a V pemutation mati and a patitioning V, and stictl positive vectos and V, such that. a) V ) V V and c) V V. Whee eithe V o V ma e vacuous, i.e. have no ows whatsoeve. oof : Fist, we shall show that VV I is othogonal. Let QVV I hen QQ (VV I) (VV I) (VV I) (VV I) 4VV VV VV VV I ()

34 () 4 I [ ] I V V Similal, we can show Q Q I Fom the main theoem, thee eist a positive vecto and sign mati S such that (VV I) S ( * ) and pe multipling this euation V, gives (V VV V ) V S ---- () [ ] I V V () S V V Now, let e a pemutation mai so that S, whee and ae stictl positive. E n () ma then e witten ( ) I S V V i.e., ( ) ( ) ( ) mati pemutation is V V V V a V V V V V Which is (a). Witing en ( * ) in patitioned fom, ( ) S I V V V V V V V V V V V V

35 V V VV VV VV ( Since V ) V V and VV heoem.5 : Let e unita. hen thee eist positive vectos and and uniue sign matices S and such that (i) S i. oof : Let ib, whee and B ae eal. Now, we shall show that is unita if an B onl if is othogonal. Suppose the mati is othogonal. We have, B ( ) ib. * hen * (ib) ( ib ) BB ib ib..() I ( BB I, B B ) Similal, * I. Now, suppose that is unita. Fom () if is unita then BB I, B B. heefoe B B B B BB B B B B BB I I ( BB I, B B ) Similal, othe eualit. Now, if is unita the theoem. gives,, S, such that 5 ()

36 B B S S () Whee >, > and S, ae signed matices. Now, (i) (ib) (i) B i ( B) fom euation ( ) we get, ( i) S i. heoem.6 (ucke s theoem) : Let e a skew smmetic mati. hen thee eists u > such that u > and u u >. oof : Since is skew smmetic then (I ) - (I ) is othogonal. So that, fom the main theoem, thee eists an > and uniue sign mati S such that ( I ) ( I ) S e multipling this euation I, ( I ) ( I ) S S S ( S) S i.e., u v. whee u S and v S. Since S is sign mati, u j j o. heefoe u >. Simila v >. But, u v u u S S ( u v) >. 6 ()

37 () 7 heoem.7 (Ke heoem) : Let n m R then thee eist n R and m R such that >,, > and >. oof : ppl ucke s to, B. So thee is u such that Bu and Bu u > Note that Bu > implies. Let. hen, cleal and >. heoem.8 (Fakas Lemma) : Let n m R and let m R oth e aita. hen eithe.

38 () 8 (a) o such that () and that such > oof : ppl ucke s to, B. hee is t u such that t t Bu fo, t t () and t t t t t t Bu u > () If t >, then the vecto in the aove sstem ma e nomalied so that t. hus () gives,. Now, let e such that <. hen

39 [, ] heefoe, can not e a solution of sstem (). If t, then () gives > and () gives >. hus is a solution of (). gain, as aove, if is such that, we get <. heefoe sstem (a) has no solution. heoem.9 (Gale s heoem) : Let R m n and let eactl one of the following two sstems has a solution. m R oth e aita. hen (a) R n such that () such that and oof : ppl ucke s to, B. We get u such that t t Bu..() and, fo 9 ()

40 () 4 ubu t t > i.e., t t >..() Now, conside the two cases : Case I : If t, fom euation (), >, which is (a). Case II : If t, fom euation (), > and. Futhe, euation () gives >. Since >, ma e scaled a positive facto so that. hee foe,, and. heoem. (Godan s heoem) : Let n m R e aita. hen eactl one of the following two sstems has a solution. (a) that such n > R () e and that such > oof : ppl ucke s to, B e e

41 hen we get u such that t and et Bu ( Wite ) e et u Bu > t e Now, conside the two cases : Case I : t, e > Case II : t, > and > and e >. heoem. (Motkin s heoem) : Let [ ] whee m n i R and m i i m n R e aita and let m. hen eithe (a) n R such that >, and o () m, and R such that i i and e >. i 4 ()

42 () 4 oof : ppl ucke s to the sstem. B e e, to get that such t u e et Bu [ ] fo e et and ()

43 () 4 u Bu e t et > Now, Case I : t,... e whee e > Cleal >. Net > and > gives. hus (a) follows. Case II : t, () gives ( ) ( ) and i i i (Wite fo ) Since t, e >. Hence, e, i i i >, which is (). ()

44 () 44 heoem. (Da s heoem) : Let n m R and n R e aita and let m d R e stictl positive ut othewise aita. hen eithe (a) o that such d d satisfing m R () n R such that d <. (whee is a vecto whose components ae i.) oof : ppl ucke s to the sstem, B d d I I d I d I to get, u t >, such that Bu d d dt dt t t >..() &

45 () 45 u Bu d d t dt dt t t >..() hee ae two cases : Case I : If t, then () gives, > and > Since t, again () d > and d >. heefoe, d < < d. which is (a). Case II : If t, () gives d d > Wite and. hen ( ( ) d ( )) > d < d <

46 Refeences. C.G. Boden, Simple algeaic poof of Fakas lemma and elated theoems, optimiation Methods and Softwae, Vol. 8, 85 99, David Batl, shot algeaic poof of the Fakas lemma, SIM Jounal of Optimiation, Vol. 9, No., 4 9, 8.. chia Da, n elementa poof of Fakas lemma, SIM Review, Vol. 9, No., 5 57, C.G. Boden, on heoems of the lte native, optimiation methods and softwae, Vol. 6,,. 5. David Batl, Fakas Lemma, othe theoems of the altenative, and linea pogamming in infinite dimensional spaces: a puel linea algeaic appoach linea and multi linea algea, Vol. 55, No. 4, 7 5, Baaaa, Sheali, Shett Nonlinea pogamming, heo and lgoithms, John Wile & Sons, Inc., Baaaa, Javis, Sheali, Linea ogamming and Netwok flows, Johnwile & Sons, W.Rudin, inciples of Mathematical nalsis, Mc Gaw Hill intenational editions, ()