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!"#$%&' "( )*"'"+*,&' -.%&/*,0!"! det -2 λ d < 0-0 2 trace - #$%&'( )* +,( -./&(0 2 3,( 3'.4(.56 6(3('7$5.53 2. 8 8 7.3'$9 2' :,$4, $30 67$5.53 ($%(5-./&( $0 07.//(' 3,.5!* +,( ;.'.</. 0(;.'.3(0 3,( '(%$50 :$3, '(./.56 47;/(9 ($%(5-./&(0* =5 3,( /$5( 6(3 = 3'.4(!>. '(./ ($%(5-./&( 4'00(0!? 5 3,( /$5( 6(3 = 3'.4(!>. '(./ ($%(5-./&( 4'00(0!>.56 2' 47;/(9 ($%(5-./&(0 4'00$5% 3,( /$5( 6(3 =! 4''(0;560 3 3,($' 76&/&0 4'00$5%!* @7( 2&'3,(' 6(3.$/0.<&3 3,( /4.3$5 2 3,( '(./ ($%(5-./&(0.'( %$-(5 $5 #$%&'( A* -2 - ( ) A = 0 θ 2 θ 0 det non-feasible region 0 λ d < 0 λ d > P() > 0 P() = 0 P() < 0 R 0 = - 2 A = ( λ θ 0 0) #$%&'( B* +,( -./&(0 2 3,( 3'.4(.56 6(3('7$5.53 3,.3 4.5 <( '(./$C(6 <D 8 8 E(0/$( 7.3'$4(0!* +,( 2'7&/.0 4''(0;56 3 3,( %(5('$4 2'7 2! 5 3,( 4&'-( $56$4.3(6 <D 3,(.'':> :$3, λ.56 F3,(G θ( i ) 2'(( ;.'.7(3('0* H3( 3,.3 P (!) =! 3'.4( + 6(3* #' E(0/$( 7.3'$4(0 ( a b c I) > P (!) =! J a J,* trace K2 :( 3,(5./0 /L :,$4, '(%$50 $5 3,( F3'.4(>6(3GM;/.5( 4.5 <( '(./$C(6 <D %(5('./ 55M 5(%.3$-( 7.3'$4(0 F#$%&'( "G> $3 <(47(0 4/(.' :,D 3,( P (!) > I 4'$3('$5 6(0 53 (93(56 3 %(5('./ 55M5(%.3$-( 8 8 7.3'$4(0* +,( 3'&</( 47(0 2'7 3,( '$%,3 &;;(' '(%$5 <(3:((5 3,( /$5( P (!) = I.56 3,( ;.'.</. 6(3 =! N 3'.4(8 $5 :,$4,./0 3,( 0(456 ($%(5-./&( $0 /.'%(' 3,.5 5(* #$%&'( "./0 0&%%(030,: 5( 7.D 03$// ;&3 3,( P (!) > I 4'$3('$5 3 %6 &0( (-(5 :,(5 3,( &56('/D$5% ;;&/.3$5 6D5.7$40 /(.60 3 7'( %(5('./ 8 8 7.3'$4(0.0 <.0$0 2' 3,( 4./4&/.3$5 2 3,( 5((6(6 $5-.0$5!35(00(0O.5D '(0$6(53> $54/&6$5%.5D ;'0;(43$-( P@@>,.0 $5-.0$5!35(00 C('* Q(54(>.5D 7&3.53 (R&./ 3 3,( '(0$6(53!560 $30(/2 5 3,( /$5( 0(%7(53 2'7 (3'.4(>6(3) = (I,!) 3 (3'.4(>6(3) = (8,!)* S( 0,.//.00&7(!'03 3,.3.3 3,( '(0$6(53 ;,(53D;( 3'.4( 8 = N 6(3* F+,$0 456$3$5 $0 %&.'.53((6 :,(5 2' 3,( '(0$6(53! $0 $''(6&4$</(*GT00&7( 5: 3,.3 3,( 03'.3(%$(0 &56(' 450$6('.3$5.'( 4,.'.43('$C(6 <D. 3'.$3 -(43' 2'7 07( R n * K2 3,( /$2(M,$03'D ;.'.7(3('0
!"#!"#"!" $%&' ()* +",%-.(/ det non-feasible region A = ( λ θ 0 λ ) -2-0 θ 2 A = ( θ 0 ) 0 λ d < 0 - λ d > 2 trace P() > 0 P() = 0 P() < 0 λ d = $%&'() "* +,-.%/%,0 ()&%2 3( - &)2)(-/ 2242)&-,%5) # # 6-,(%7!* 89) 3(6'/-: ;(():<2=,,9) &)2)(%; 3(6 3! 2,9) ;'(5) %2=%;-,)=.0,9) -((>? >%,9 λ -2= @,9)A θ( i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det 0 non-feasible region 0 0 λ d < 2 trace P λ d > ( resident) λ $%&'() I* J;-,%2 3,9) ;9-(-;,)(%:,%; </026%-/ %2 =)<)2=)2;) 2,(-;) -2= =),*
!"#$%&' "( )*"'"+*,&' -.%&/*,0!"# $%&'()*+,(&- &'%)*. = / 0*& 2+ %3 *4%$56*7 89: $%8 &-(3; 9< % 595:6%&(93 6(=(3> (3 &,9 5%&)-*+,(&- &-* +&'%&*>8 5%'%$*&*'?*(3> &-* $(>'%&(93 '%&*?*&,**3 &-9+* 5%&)-*+ %30 39 $(>'%&(93 <9' &-* 5'9+5*)&(=* @AA B3 &-%& )%+*7 &-* 5'*=(9:+ %'>:$*3& >9*+ &-'9:>- 93 &-* )930(&(93 &-%& )69+* &9 &-* 5'9+5*)&(=* @AA 0*& <!7 9' *C:(=%6*3&68 &'%)* <. B3 &-* <9669,(3> +*)&(937,* +-%66 *4&*30 &-* %?9=* '*+:6& <9'.. $%&'()*+ &9 >*3*'%6 n n $%&'()*+ 2+ % 5'*5%'%&(937 D(>:'* E +-9,+ $9'* 0*&%(6+ 9< -9, (3 &-*.. )%+* &-* 69)%&(93 9< &-* )-%'%)&*'(+&() 596839$(%67 %30 -*3)* 9< (&+ '99&+7 )-%3>*+,(&- &'%)* %30 0*&!" #$% P(&) > ' ()*+%)*,-.,) /%-%)0 n n 20+)*(%3 F*& P (λ; Y X) := 0*&(λ4 5(Y X)) 0*39&* &-* )-%'%)&*'(+&() 596839$(%6 9< &-* 3*4&G>*3*'%&(93 $%&'(4 9< &-* 6(3*%'(H*0 083%$()+ 9< % Y $:&%3& (3 %3 X '*+(0*3& )9$$:3(&8 I*45'*++*0 +9 &-%& &-* 6*%0(3> )9*<!)(*3& (+ 59+(&(=*J Theorem 23 $30*43%5 6 *0 #%*%7&4&8'3 *( (i) 523 43/"+$&92*, 9&$&/353$0 4393%4,"%5*%#"#0'. "% 523 5$&*5, (ii) 523 5$&*5 09&,3 *0,"%%3,534, (iii) P (!; Y X) > K f or all Y = X, &%4 I(=J ( P / λ)(!; X X) = K (:2*,2 *0 &':&.0 523,&03 *(!(X X) *0 *$$34#,*8'3) "$ ( P / λ)(!; X X) = K &%4 R K <! ("$ Y,'"03 5" X (523 '&553$,"%4*5*"% *0 +#&$&%5334, ("$ 3;&/9'3, 8. (. P / λ. )(!; X X) = K &%4 ( P / λ)(!; Y X) > K ("$ Y,'"03 5" X). <% 523 "523$ 2&%4 *( 523% 52&5 Y,&% *%7&43 X. P (!; Y X) < K f or some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λ >! +:)- &-%& P (λ; Y X) = K A(3)* &-(+ λ (+ %3 *(>*3=%6:* %30 +(3)* R K (Y X) (+ 09$(3%3&7 *(&-*' R K (Y X) = λ >! 9' R K (Y X) > λ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
!"#!"#"!" $%&' ()* +",%-.(/ Y = X: other Y: P λ not allowed by assumption: impossible: $%&'() *+,-.(-%/& 0(2. 3%-'.-%/ 45)() -5) 62%/./- )%&)/7.8') )9'.83!: ;)(-.%/ <%0'(;.-%/3 0-5) (- =.--)(/ 0-5) ;5.(.;-)(%3-%; =8>/2%.8 0. =3%-%7) 2.-(%?.() )?;8'6)6 <> -5).33'2=-%/ -5.- P @!A Y X) > B+ C/ -5) /)?- 3);-%/: 4) 35.88 ;/3%6)( %/ 2() 6)-.%8-5) -8<? 0( 5%&5)( 6%2)/3%/.8 -(.%- 3=.;)3: 45%;5 %3 <.3)6 8.(&)8> / 8;.8 ;.8;'8.-%/3+ D)(): 4) /8> 6%3;'33-5) %2=8%;.-%/3 0-5).<7) -5)()2 0( -5) 6(.4%/& 0 ECE3+ $( -5) ;.8;'8.-%/ 0 ECE3 /) ;./ 8.(&)8> ;/;)/-(.-) / 387%/& 0( Y %/ P (!; Y X) = B @$%&'()!BF+ C0 /): %/.66%-%/: G/43 32) =%/- %/ -5) @X: Y FH=8./) 0( 45%;5-5) %/7.3%/!-/)33 %3 /)&.-%7): -5)/ -5) %/7.3%/!-/)33 %3.83 /)&.-%7) %/.88 =%/-3-5.- ;//);- - -5%3 =%/- <>. ;/-%/''3 =.-5-5.- 6)3 /- ;(33 /) 0-5) 38'-%/ ;'(7)3 0 P (!; Y X) = B+ I3 3/.3 3';5. ;'(7) %3 ;(33)6: -5)!-/)33 ;5./&)3 0(2 /)&.-%7) - =3%-%7)+,%2%8.(8>: %0 /) G/43. =%/- 0( 45%;5 %/7.3%/!-/)33 %3 =3%-%7): -5)/ %/7.3%/!-/)33 %3.83 =3%-%7) %/.88 =%/-3-5.- ;//);- - -5%3 =%/- <>. ;/-%/''3 =.-5-5.- 6)3 /- ;(33 /) 0-5) 38'-%/ ;'(7)3 0 P (!; Y X) = B+,%/;) %/ -5%3 ;.3).83 32) -5)( )%&)/7.8') 2.> =.33-5('&5! 45)/. 38'-%/ ;'(7) %3 ;(33)6:!-/)33 2.> ( 2.> /- ;5./&) 3%&/ -5)()+ since: Im i Re + +? P(;Y X) = 0 $%&'()!B+ J5) %/0(2.-%/ &%7)/ <> -5) 38'-%/3 0-5) )9'.-%/ P (!; Y X) = B %/ -5) (X, Y )H=8./).<'- -5) 3%&/3 0-5) %/7.3%/!-/)33+
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s(y X) := ρ(y E *,,) (X)). @-2(,4 2(,)*2, 46*& B%&)& [ s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λ; Y X) := 5&,(λ- #(Y X))8 *(5 /-) Y 4:/!2&(,+. +-4&,- X8 [ ] P (R M (Y X); Y X) > M, λ s(y X) +((R M(Y X)), T ) (X X) 2( 6*),2:+*) [ ] s s (X X) = Y X (X X) = RM / Y (X X) T ) *(58 B%&( ( s/ Y )(X X) = M, Q s Y Y $ (X X) = [ Q R M /( Y Y $ ) T ) ] Q s (X X), X X (X X) = $ [ Q R M /( X X $ ) T ) ] (X X) [ B2,% T ) (Y X),%& >&*( *'& *, )&6)-5:,2-( -/ Y L2(52725:*+4 2( E *,,) (X)8 T ) (Y X) = M a. $ (a)/ 5a ] (Y X)8 [ (a)](y X),%& >*,)2K -/ *7&)*'& 6&) *62,* 92),% )*,&4 *, *'& a -/ * Y L2(52725:*+ 2( E *,,) (X) 52//&)&(,2*,&5 *-)52(',- 92),% 4,*,&48 *(5 U *(5 V,%& )2'%, *(5 +&/, &2'&(7&,-)4 -/,%& (&K,L'&(&)*,2-( >*,)2K # = M (a) 5a8 -L(-)>*+2D&5 4:%,%*,! $ U =!8 V $ U =! F!NG<
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