Radici p-esime di matrici: proprietà e metodi numerici
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(41) 4 p = =9 = ] 3 4 A k ( 3T3 7 87 54j= = 3 4 k 6( / / pF !" # $ % WV V 64 r k A WV( ` T(mc3 r( 6H / kX= (r '#&kp F & n×n. p. 1/p. • (A1/p )p = A • A1/p. '. V9 / = 1 )3 54 =63 pF ( R=9= )4 & Sp = {z ∈ C, z 6= 0 : −π/p < argz < π/p}.. ( 4 k 6( r7 4 rT79)3 = 64 6(9( =
(42) 8()(&& 53#=94 kr6(9( )4 63 N3 r 53 M / p&rF ' { { >=?4 k* / N3M3 * / &r ' r( R=9= )4 T33 5H' )4 E= r3 6(j7 'k FF &r ) 3 W V = 64m1'k H ' 53E= )4 r 3T3 H'H'64 = r 54 kk 'X=?4 )3T3 t 3{) 3 k5 3 633 5H)4 E= / 9 N=?4 r A ( 7 /,+ !>=9= (j1' 53#H'64 t ` c7 - rr = H54 N &k ) 3T3 kX = E =?Z log(A) = p log(A ) t 3 7!4 5 ' 3 k =R = 4 9)4 Q34 r pF 6( *7 4 rT79)3
(43) / 9 >=?4 k A = r =9/= =9=| ) 3E4=?4 WkV x ' ( 9 ) 3 1! # (j= > <=9= <0 (j/=94 =j6 = 4 ) ( ` kX4 # = (j 1') r3#5 = 3# 36 = H'kN 5 = 4 r) 3#3 r V > 4=9T= ( ) 6 4 4mT 1& &N &kt 3<r& ' F q k 4 F o
(44) 5 7!E p3F 7 5 3 ' / 4 r = F (z) = z ((1 + z) A − (1 − z) I) r `. -rkX = N =94 kN 3 ' k + 7!3 W V
(45) ) H' ) 3TH 5 4 E = 6 ( T (j= kX = 7 6 4N 3k) 3 ) 4 32 N =j= 5 4 T &&) &r ' q k 4 F o&5 7!^7 /,+ (9( 4 / = T 3 # &6&kN = 2 7 6 4RN 3k) 3 ) 4 A ( ( r5 7 6 3#3 5 (m=?4 c = 6 ( (m= N = 9 / 9 79N 4j= 4 kr5 H53 4 * 7 4 ( r = N 4 ` 5 4jH kx 3 2 7 4 5 7!4 R =WZ 7/ 4 # 3 R 1 X = 6 3#3 4 k pF ( / 9 F =94 r 9) 3#3 ) 3E=94 ( 4 # 1' 4 9) 3 T &&) 4 tr '! 4 = N 4 t W Vtk = / 7 3 rkX = ) &r R (97 4 r = N &k '& / 4 7 4 rT 79N 3 = ` 6 ( # (j= k = 3 =9= 6 4 N = / 4 l 7 6 4R# 3{k) 3r5 3 6 3#3 2 4 r pF ( / 9 N =?4 k 0 ) 4j= k) 3 ) 4 N =9= k &r ' (j= N = x 4 # 1'5 3E= 1 / (j= '2 / ) 3 3 r ' 7!3 (9( E =WZ k) 3r5 3 T 3= r 7 {6 ( r / &k ' 3 2 (j= ) 'T 3 E =WZ / 4 3l 4 ) ' / (j= 6 && = ( ( / E 1# ( 0 / N =j=?4 ON 7# = 5 33 & 6 3z7 4 5 7 | 1'6 49> = O6 3#3 R %9# &k ' '9) ( ( X =?4 ` / k# 3 k kR =9= z 4 k N =94 r 8 ( F ( 6 4 E 1' '2 3 7 4 k# 79) 3 e 7 4 5 7 4 =?Z%9 tN 494 E 1) 4 N 3T3 6 3) (9( # %9k) &r '
(46) = / =j= 3 4 r pF 6 ( / 9 N =?4 r 'G ( H'5 3) 4 1/p. −p/2. p. p. ;. 1/p.
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(48) P 6 3 ( kr ' k)7#= 53 c1`r*7 4 6( r = N= T3E R= ` P Rgh= ' )7 7 3 kN= )3 F 3 . / )&k ' p N3T3 3 / r k7 4 5' 3 k{(j= N'T3 E=WZ = )3= ` 1'r H' ' X=?4X ` )−=9= A =63T03 Q1 )4 = (m= )'#3 h7 4 1 / = & &o 5(9p !( q N3#= ,) rXH N4 X= T( k WV ( 3 [ N=?4 k A ( P gh= '=?4O r '
(49) '1 499% 'H E= 7 5( #= E 1 / =9 = )3T`1 3 54 r 3 / ((j/= kr6(9( = ' ` 'H r 'G4 4 NT( = ) 3#'1 9)3##3 3 E7!= 4 54 N' k& 3 k ' 3T3 (j= )' #3 Y=WZ E3 )( 8A'H 1/pk4 )3 s k H ' / M7 4 ( kX= >= 63T3 ( 53 / r& ' ( #'H E9% >= E'1 1 / = t 9)&k& ,D" V k 'X'1 649H ' / !4 N= k r = 7
(50) 4 5'H ( N=?4 r A 7 4 #1 / = 1 )3 54 4 k)3 { G7 )( #= #`1 t !9% 1 k [)7 7 3 kN= &T3 = ` P gh= N3T3 / Nk& ' X −p −A = 0 ' 7 64 )=j= r64 A1/p 14 r / )3 k 53 )4 z3 '1 49(
(51) 3!7 / = Q% (9( 63T3 / k& ' WV *( ( )=9= r )3 R= ` 8( M4 7 64 + (!(j= N' T3 E=? Z / 64 k$&z7 4 / 6(j= (j= N= k ( 64 N= / 9c( / c1 )4 = V )' ' )(j=?4 N= 6(j( 4 *(j= )' #3 P 63 = 64jk& N7 #= 53 '1 k!'H ' 6( 64 #=9= r R= ` WV ' ( 9' )( ( / E= 64 )k& ' / k& '9)3 / = #3 #&& = T7 53 5H #'1 49( E9% kN= 0WV / ( 4 N7 7 4 ( r = )k& '.X= H54 )3 =?4 N(jh 54 N= i 3 N=9= 54 #&6k& )k& ' q r4 F o& 57! 1'64j( ' ~7 53 ( / 4 kX=tr `.-k kX= N=94 kN3 (j=94 N= 6H' M1 )3 / = )r& ' = 6497 )3 )r& ' kE `{ . / 4 4 V ;!" V 9' )( N= ( / 3 k)3 r t 3 7!4 #'3 = ` &U WV / 4 7 4 57 )(j= &9 U E=? 3 &63#3 hh )4 R ' &DUWV / 4 k / )3#H' 54 Y= 7 )3#= h4 5' / (j=
(52) k.-rr = F WV 7 4 + 9V c3 r ' '1 rr = E'1 4 / Or 5(j= k '87 / = )k& '9)3 3 R1 N= s k H ' Q7 '7 4 6( r = N= / R= ` 9' )( >= ( / 3#3 4 N7 7 4 ( r = Nk& '< = 6H54 )3 A1/p '1 r3 X= H54 N3 R9% E= l( / 3#3 rT4 r '!h 4 k!& / E= )4 63 7 k ' F 7 3 6(j( `1`r[)3 k 53N= k = `3 h )4 / 3
(53) kD=?4 )7 6&r!H'k 64 N3T&&>= ( / ` . / 4 4 ' / 9' )( >= ( / 3#3 c4 / k& 'N3 N3 k )3 3T3 l / r& *( 'H / 57!7 54j= / 9 N=?4 r AF rT4 r 53 = [9 54 np ' / = ` V 4 / k7T3!7 4 5'!3 F t63 N3r 53 t A1/p )3#3 N=9= 54 #&6&k)&k ' ~q k4 F o
(54) 57!< / {7 53 ' ( / 4 r = r `.-k kX= . N=?4 r)3 !% / R= ` V 5(j=94 T3^3 H =?4 A1/p 3 '1 49( '7 / 7 )3 ' ( / 4 kX[= k `.-rkX= N=94 kN3 !7 4 #23 / )3 c( )7!7 3 = rk WV 1 )3 / = )r& ' = 6497 )3 )r& ' G3 #= 64 )k& ' f 4 = t 3 / N4j= k)7 E= 53 k N= )3#3 c7 c(j7 64 kX= )r& ' / 4 8(j'1 )3#= WV 9V 7 64 (9( Pr '! 4 ' = )4 E'1 64j( R= ` ( / 3#3 9' )( 63#3 (j= N' T3 #=?Z / 4 4 5' / (j= &6k& r '87 / = Nk& '9)3 r ' 7!3 (9( E=WZ t 97 )4j= r 53 N4 )' ' a( 63 6r& 'N= / v ( k < N=94 k~= (j= )= N= k)4 N=j= 4 T(m= V (97 rkE9% V N=9= O '1 64 E9% N4 a3 - r kh1 )4 R= ` U ' (j= N= ! >=9= {k '!( 64 >= N=?4 rEr ' / = 1 )3 )4 ' r r '#k& '9N= N=?4 rE F 7 rkX= =9 / 79)4 =?4 k t / = 1)3 54 )4j'#=94 )4 r =
(55) )3 k ' T&r '9>= N=?4 rk ' / = 1 N3 54 V l( ' 0)4j' #=94 )4 r = `1 k . F " )y3 64 6( r64 63T3 r ( ' @.
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(57) . / =9= R= `( 4 #=j= 3T3 = ( rkk&k ' 63 R= ` FU#=9V ( ' (j= >= 7 3 rkX= N= 3 H / )H5H . 54j=94 k5 z7 4 5H)4 r '87T3N= r 53=k '87#3 >= )4 ()V .0/ E=?( /xkXK ( ' l(j= N= =( 6H / #= ( / / 9g< 549p (m= N= ' r ' 7 / N=?V 3 ' "J5 ) r ' ( T(j= k 23 /M P 63#3 = ( ( c4 T7 54m= N= k X= H54 >%9k `4 #( / 3E= N= 7 ( TH'E%9>= E1 )=9= k / = < = 64 <64j4 54 x4 ( / = k87 [k)3r 53 ]i )( / P= ` (j= N= r '! 4 ' = N= k ' 3 87!3 kr = )&k ' 63 = ` UE=?V 9N= 9N3T3 * / &r ' 0 C >=W()' $ `4 T( / 3#= N= 3T3 G(97 64 kX= )&r ' k49( WV R= `4 krkX= kkX= =?4 ` )=j= [ ; ')( N= ( / = rk WV |1 &kN= / )3 . . N=j= 54 T&&)&r ' [ q k4 F o
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(59) 7 53 ' ( / 4 r = ( ' 7 4 7 4 5' 3r9 F (m= )'#3E=?Z& / 4
(60) <D4 )' / (j= 6&&r7 4. / T33 54
(61) X= 4 6(j( 4 z7 4 R=9= r = = k 54 k k / a7!4 5' 3 k )7 64m= 83l7 )(9( #'T3 E=WZ / 9c3 54 l(j= )'#3#&6&k)&r ' t = ` '9N( N= ( / 3#3 E= 64 )&k '& P gh= ' X= H54 N=
(62) 9 / 9F9)3#( & (m= )'#3E=?Z ( ( ' 1'kr 4 E1'63 >= ~79)4j= r 53 N4 k = . - k k k]V9 5(m=?4 >= / M )=j= c(j= )'#3E=WZ 63#3 )H)H' 54 7)4j= r = 6(j= =R=9= / N= WV( N3 / k)( V9 5(j=94 N= / )3WVh7 4 5' 3 k 4 5' / (m= &6&
(63) Q/ 6(m= = ` !V9 T3 H54 1 = )H5H' =1'4 / k )(j= r ' 7 / = )&r 9)3 & V 64 ( rk ' log p ~t / (j= 63)(9( = ` 3 E= 4 )&r '& P gh= tWV 57 4 / 9( )3N= / 4 ( / 3T3 N=?4 r79)4m= r &c( 5(j=94 N= l4 5' / (m= (j= )'#3x= / =9= EM= (j=
(64) =9= / N= t 37 R= ` U#=9V / 63T3 WVGV9 )(j=?4 N= #H53 544 5' / (j= &6&k V ( 63 k)(
(65) >=?4 kRr ' / = 1)3 54 R N3&k T&r 9N= 3a( / 7 4 6(j= )&k ' ( ' 4 #( / 3E= N= 3H5H'4 k = Dh64 54
(66) / 63#3 3T3 )3#H 54 E= P gh= ' r ' ( )3N= / 4 t 3 7!4 kT7)3 3 Y= t3
(67) R= ` U WV / 4 4 G3 3R1N= r 5(m= r '87 / = N&k 9)3 7 V
(68) 64 ( r k ' p2 &87 k#( r 3 ( / )7!7 3 )&r h7 64 1) 3 54 3 R1 N= { p. K.
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(71) W4L0NOTY@CX }B{TAW H. t / 6 (j= 7 4 )7 #= )3 57 |1'64RN= 63#3R%9#&k = '9)( 79)4 )H54 Nh !E" =?4 ` / 4j4 k T3
(72) k ' kR=9= F / &r ' N=94 r79)4 )H54 Nh !<; r r 'rk =94 4 k 7)4j= k )3 )4 ( / 3#34 ky N=94 k y / . 6( 4 E1'64 k G3 7 4 rT97 N3 7 4 57 4 =?Z 637)4 )H)4 Nh ! @ i 3 )(j( #% V 64 r = / =9= 3 4 k pF 6( / F N=?4 r ' ( H 53 N4 9N4 k r '#k& ' ( / - rr = `7 43 6( T(j= k k& = N3 4 r WV 63 k)( N=?4 rM( 'H )3 )4 t !9% t =94 ` / 494 r 6H53 (m=?4 / r = / = T3 7 4 7k)7 E= 53 ( / rk6(j( E`1 / )3 #3 r 'r=9= 54 N=?4 r)3 3 57 64 N= )4 2'1 k T3 7 4 ` )=9= 0r4 'r p 4 3T3 57 7 )4j= / r N=9= 54 #&6k& )k& ' E N=?4 r.
(73) . !". U 4 T(j7 =j= #1 r = T3 k 7 0r / 4 r '87 3(9( #3 87 0k / 4 C4 k)R3 i ' C R WV a4 T(97 R=9= #1 r = 3 ( k63#3 N=?4 r" / !4 N= { 54 n t63rk = {k '87 3 6(j( E t63 kkX= 4 k)3 ( N =?4 k 1'494 kN= r ' 3 3R=9= 4 t / ( k 53 r =94 3 )4 03 F kX = e ( ) 4 N= 9k ' 3 3=j= 4 / ( k )3
(74) k = #%9kN= 9 / r#hT3 7 4 l 4 N 7 7 4 ( r = = * 3l 4 T H T 3M( kr ' 3 r 53 9 3 r ( 6 H / kX = () = ) &k '3 & $ ( 4 r 8 $ N = A ∈ C ( ` %9# ( r< N =94 k& % (' % )' ! +*-,
(75) % A 3 & N=94 k • = N 3W V & B∈C n×n. n×n. m×n. n×m. 1' a¯ #3Er ' / HN= 63( / 64 r ' 7!3(9( a )(9( T3{ / 4 k ' F 7 3 (9( |1'kX= (j= 6(j( 7)4j= 4 k)3 a 79)4j= NH' 9)4 57 7 )(j= N=?4 k =94 )(j7 5(j= k ' / H>= A 1'494?Z kN= k 'G3 23R=9= 64 A U A ∈ R 3 =?4 )(97 )(j= vr ' / HN= A k ' r r ' 3 N=94 k % (' % %9E = k 'r 3 N =94 k B ∈ R = )3WV& bij = a ¯ji. ji. ji. ji. H. m×n. n×m. bij = aji .. +.
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(77) >=?4 k =94 )(j7 5(j= . 1'4949Z N= k 'G3l3R=9= 64 AT ( A = AH *P 6y3 N( 4 )3 A ∈ Rn×n =9= . A. % ! A = A ! ' % ( AA = A A • A∈C * ! % ( A A = I P 63 k)( r4 )3 • A ∈ C ( n×n • A ( ∈ C=?4 (. T. n×n. H. n×n. H. H. AT A = I. A ∈ Rn×n. ' % ' , ' ! % . 7 4 % ! , ' % * ' ! ' ( r6y }
(78) $ >=?4 k =?4 !H' 53N4 Q( / 7 64 54 aij =3T3 0h 54 i >& j (i < j). • A ∈ Cn×n. . ? ? ··· ? ··· A= 0. ? ? . ?. ! , * '
(79) ! ' ( a = 0 7 64 i > j + 1 (c( 6H / k = N=94 r xh 54 o
(80) (9( r' 649H ( / 7 64 )4 &. • A ∈ Cn×n (i < j − 1). r6y }
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(82) (9( k' 64jH *$ % ! , ' % * ' ( / 7 64 )4 Dh64 54 [ ' 7 )(9( #'T3 <WV7 / 763 kkX= k '!( k / = E1`9 63T3 ( )=j= ( 57 4 )H' 9)3 ( 7E1'649( '9h&k4 i + ( #H'E%9[WV~3 [ N=94 r ( / 7 6794 N4j )= 4 # &k '9!Nh6= 4 5 4 ' c3 `' r3 `Vk
(83) WV r ( ' )3 7 2 × 2 =?4 H' )3)4 A r6y }
(84) (l( 6H / kX= N=?4 r h 54 / )( F =?4 !H' 53 )4 ( / 7 4 54 &. • A ∈ Cn×n. . A= . ? 0 0 0 0 0 0. ? ? ? 0 0 0 0. O. ? ? ? 0 0 0 0. ? ? ? ? ? 0 0. ? ? ? ? ? 0 0. ? ? ? ? ? ? ?. ? ? ? ? ? ? ?. . . .
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(86) • A = [aij ]. / N=94 k ' % %8( . 7 5 ( #= #1' r6y }
(87) $ >=?4 k `7 3E=?& 63T3h 54 $& aij = ai+k,j+k . a0. a−1 . . . a−n+1. a A= 1 . a0 a1 .... an−1. •. ∀k. a0 a1. a−1 a0. . . . V A r ' 7!3= kX= %9E= )3T3 ( / Q7 4 4 TH'
(88) <9)3#3 L (j( 4j1` ( / c7 4 k 53 '9 1' c = [c0, . . . , cm−1]T f / ( k)3 )4 #1'49( Af (c) = [aij ] k& 64 / 9 N=94 r f )' % ! % E k ( ' m × n ( & 7 4 i ≥ j ci−j mod m 7 4 i ≤ j aij = f ci−j mod m r6y }
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(90) 3 )( m = n = 4 c = [c , c , c , c ]T )=9= k {3 0 1 2 3 ( 6H / kX= N=94 k f F k#4 k )3 = & c0 f c 3 f c 2 f c 1 c1 c0 f c 3 f c 2 Af (c) = c2 c1 c0 f c 3 . c3 c2 c1 c0 . L (j( 4j1` WV A (c) 79)4m= k 53)4 / 9 N=94 r0 `67!3#=9&Or k '87 3 R= r = R%9fE= )T3 32( / c7 4 k 53 '9 c 9)3#3 2( N3 )4 f $ >= A ∈ Cn×n, λ ∈ C x ∈ Cn '0 / 3#3 = N3 =WV & Ax = λx. . ( {#4?ZWV λ / -* % ' % ' A WV x % -* % ' % % ' r 5494 T(97 kX= λ !( r H53 / = 1)3 54 . / 90 >=?4 k A WV r )= 4 r Or ' Λ(A) r 5(j= #= / T( r83 % % ' A #3 ` / 3 t N(9( ρ(A) H53 / = 1)3 )4 . A . r &. ρ(A) = max{|λ| : det(A − λI) = 0},. RU=9 = ) , , ' / % % )4 % 4 )3 (A / / 4 r '87 3(9( 1'r ' x, y (x, y) = x + iy
(91) )' ' kN= 23 / #=?Z NH' 9)4 (i = −1) k. 2.
(92)
(93) L H 7 / = 3 7 7 /,+ 6(j( 64 4 )7!7 4 ( kX= N= k = 3tr ` 54 9N= 7 53)4 (r, θ) U` θ / O / 64 84 k)3 zi ' ( 4 l3 6(97!4 6(j( ' P )= WV U. eiθ = cos θ + i sin θ.. / 0 / 4 k 7 3 6 (j( E ` / 3 . eiθ. r=. (r, θ). p. x2 + y 2 .. ( ' 83 k ` )4 9N= r7 )3 )4 =637 / = x = r cos θ. Q/ . (x, y). 1. 0 ' . 37 )3T3 54 . y = r sin θ.
(94). ~ 6(97!4 6(j( ' t E3 / 64 . x + iy = r cos θ + ir sin θ = reiθ .. V R =9= Nc= 3 ' (% ' % 3E( /4 #1`6 4 r '87 3(9( , ' ! ' z . reiθ θ. x + iy. θ = argz.. .
(95)
(96) &" U S / ( k& / 4 k '87 3 6(9( $ )7 7!3k)&r '&WV 5H 63 kk = S N(9( `r / O / 64 r ' 7!3(9( =j= * ! ' !" % ' )' % 7 ( r87 3 kkr = * ! ' !" V r )= 4 r r ' U. . . . f : S → C.. z. 1'. / O 3r r = S 7 5(9( c( 64 E1'64 u(z). . f (z) = u(z) + iv(z). v(z). ( / 64 4 )3 7 64R / z → u(z). . z → v(z). ( 2 / &r =1 )3 )4 63=k 7 24 N3 i V 64 r 3 $ % , ! f `U . u. 3. z = x + iy. $ % ) % f . v. r ' x, y 4 k)3 3T3 54 #3!1 )3 54 7WV 3 / &r ' f ) (9( / R z 7 /,+ (9( 64 ( 4 #=j= 3T3h 54 f (z) = f (x + iy) = u(x, y) + iv(x, y). " .
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(99) U n / = 4 a7 5( #= E1' Gi ' ( 4 G3 / &r ' 7 )= r & 6 ( $& nF f (z) = z n .. t 0k ` )4 9N= Q7 )3 )4 M7 5(9( l( 64 #1'4 . z = reiθ. / . f (z) = r n einθ = rn (cos nθ + i sin nθ).. U D = {z ∈ C = |z| ≤ 1} U z / 063rk = D )3T3 54 z n r 54 / 3r r = D / z → zn )7 79 D ()( U S T3M( =j= 54 { / 64 Er '87 3(9( reiθ k ' 0 ≤ θ < 2π/n r ≥ 0 2 / r& ' M1)4 )'#3 Q4 )3 )7 7923 = 64m1)3#3 . r → rn. [0, 1]. ( ( 3 / &r ' . θ → nθ. )7 7923 = 64m1)3#3 C )3T3 54 3 / &r ' 92= / =j= {E / 4 . [0, 2π/n] → [0, 2π].. f (z) = z n. N7 79#3D( =j= 54 S 3T( k / #= )4 h 54 N= w = teiϕ. r ' 0 ≤ t ≤ 1 0 ≤ ϕ ≤ 2π $ N= / / 4 k 7 3 6(j( z = re 1' )H53 0 6( 4 E1'64 = / =9= / 4 r '87 3(9( w = )3 = V w = z 0 iθ. n. w = teiϕ .. 3T3 54 . wn = tn einϕ = z = reiθ ,. 9 / ( H / & . tn = r einϕ = eiθ ⇔ inϕ = iθ + 2πik.. 1' k / O = 64 B T( 53#1'r . ϕ. ϕ=. =?4 1` . θ 2πk + . n n. "5".
(100)
(101) t / 4 ζk = eiθ/n e2πik/n ,. k = 0, 1, 2, . . . , n − 1. ( = / 9= = #(j= = 7 5(9( ' (9( 4 1`T(m= k 81'64j= r 637 5 3#H' ' a4 H' 53)4 r ' n 3 N= = ( 64 E=9= 6E3 k4 WV / #= )4 k / 1'4j= k 37 / = eiθ/n L H { / 64 r ' 7!3 6(j( ω k = e2πik/n = cos. 2πk 2πk + i sin , n n. R=9=
(102) n %% * ! % / X= . k = 0, 1, . . . , n − 1,. . t 7 / X= . ω kn = e2πik = 1.. ζk. ( ' )3T3 54 #3 7!4 ` )=j= . eiθ/n. k 'G3r4 k nF 6( 3T3 / #=WZ &. ζk = eiθ/n ω k .. ' ' 1`T(m= / Vc3 / &k ' f (z) = z )(9( `k t 5H ~7 / = z / / r 7 / = w / (m= 6(j( 7 / X= w 3{ / &k ' z = √w r5 494 T(97 64 n 7 / X= {#(j= = 63 7 r7 64 / (j= T4 r WVr3 * / &k ' n. 0. z=. √ n. w. * % % -' ). 0. ,5, . 0. n. x
(103)
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(105) [ U w = f (z) / 9G / &k ' 9N3#= kt / ' D WV )7!7 3 T3[ D ( / / } G = N3 WVG3 / &k 't 1'49( z = ( / 3E= E1' ` 63[ ' G Ut 637 G 6( T(j= ' 3 / &k ' ϕ(w) / #'1 `WV 9)3 #= WV z = ϕ (w) z = ϕ (w) . . . 7 64[ / M3* / &k ' 9N= / 9 / !k& X'1 649( 3 8 / &r '1 ϕ (w), ϕ2 (w), . . . ( < k ' )3#3 )4 * !, -' %% * ! ' !" ϕ(w) % #= EE3 1 '2 G . . . .
(106). / &r / 9 N=94 k A ∈ C 1' 5H53 R%9#4 3 $ > = / 9c )&k / &r ' f{: C N→=94 Ck ( 7 4 3 ( #= / )&k 'z7 *( k87 3 r / f (z) ( / 7 )3 ' t / (j= k)( 7 )(9( ( k87 3 kr r = *( )(j= E= / #4 3 N=?4 r 63#3 )49H 'r = ( k) 3) 4 3T3 h5 4 / 3 7 64 f (z) t !N=9= 5H'X7 53 h( )3)4 p(z) = X α z , H'r6 4 # 3h7 5 3 R N =?4 r[k r `.-k k = ( )3)4 7 )=j= r / = {( 5(m= #= / k h5 4 N 3 kX = A N 37 ) (j= z & n×n. k. i. i=0. i. i. p(A) =. k X i=0. ",. αi A i .. i.
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(109) ) 3#3 )4 7 5(j( R%9T4 f (A) = q(A)−1p(A) ( lr& 64 37 53 ' q(z) A r6y }^
(110) i '!( 64 l3 / &k 'r4 )&r 9)3 f (z) = 1 + z 2 . 1−z 3T3 54 3 / r& 'E N=94 rk 54j4 #(97 ' k = & f (A) = (I − A)−1 (I + A2 ),. WV %9E= (9( WV 1 ∈/ Λ(A) 0 / H k4 )3 7 64Q / &r 9)3 #= WV( / / #( r 0)7 4j= tk = rr = H53 { / = 1) 3 5 4 = E 7 5(9( %9f#4 32 / !&k { N=94 k f (A) 7 64R6&&k 3T32 ( 6 4 3) 4R f 3#= 6 4 9N = E 1 r = N = / 9* / &k ' f (z) )3E= )3T3 = 64 / k ' F = 5 4 W V / ( Γ W Vtr ' = rx 3 !( r Λ(A) 6H53 [ / = 1 )3 )4 A T3z1)3 54 ( Q 7 /,+ %9# 4 / ( PT 3 = k5 4 k v = H54 ) 3 i / W V r = 3 fh5 (A) 4 / 3 & 1 f (A) = 2πi. Z. f (z)(zI − A)−1 dz,. 1'Q3 = H54 N3 & / 9 N=?4 r 9N= 9)3#3 >=?4 k& V QV9k ' 3r r = H)3 = H54 N 3 6 H)3{ 3r r = { 63T3 >=?4 k = H54 9 ( e7 /,+ !%&%T4
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(122) u U 7 53 . A ∈ Cn×n. $ R%9 ( '. % ! ' ' ) % % % )' A T3. P (λ) = det(A − λI) = a0 λn + a1 λn−1 + . . . + an .. w 0y]
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(124) * (' % ! ' ' ) % % % )' %% ' . . A ∈ Cn×n. . P (λ). % *('. P (A) = 0.. $ ) 3Q= r5 4 r , ( H / rN= k = WV / 3#3 T3 ( / 7 53 ' k)4 N=j= 4 T(m= k P (λ) 5H E)3#=94 l7 53 ' E / AP (λ) ( 2N=j= 54 x
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(128) " ψ(λ) H54 r= )3 &WV ψ(A) = 0 L (j( 4j1` W V# 37 5 3 ' ψ(λ) / a>=9= )4 P (λ) ( 5H ) 3E=?4 l 7 5 3 ' p(λ) W V ( / 3T3N = 9 A t ! N =9= ( / 7 7 V p(λ) = ψ(λ)q(λ)+r(λ) 1' r(λ) / l7 )3 ' r 'G H54 5 4 6 3H54 ψ(λ) ( / 3#3 N = 9 A C ) 3T3 5 4 9 & . . ( ^V V 7 'WV,( ψ(A) = 0 rk6(j( )4 kX= r(A) = 0 r(9( k ψ(λ) 7 64 R9% #&k #3X7 53 7 H)4 & & / 3#3 >= R A r(λ) r = k r = / 3T3 $ )[3 'kX= WV ψ(λ) / >=9= )4 P (λ) ( 6H / VxH53 &k64 [ ψ(λ) ( / = 1 N3 54 = A / {&r64 s r'1 49( 5'H R / = 1 )3 54 t t !>=9= h( / / = 1 )3 )4 A ψ(µ) / / = 1 )A3 54 ψ(A) kψ(λ) (9( k ψ(A) = 0µ ( H / WV ψ(µ) = 0 ~s )3 r7 4 r + T3M( 6H / kX= & y A ∈ Cn×n ψ(λ) % * ' (' % ! ' ' ! ' *
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(132) t 1 )3 )4 . ( ' . . % ' %% * ! ' !" * %% ' ' % % ' t ' ' E k ! % * y%% ' % % )' r r < ` k r k r ' 9 ' r ` k 9 0 ' 9 9 ( 9 / / r ' k ` / ' ` r . f (j) (λi ) 0 ≤ j ≤ ni − 1, i = 1, . . . , s ( ( # j ( = ( 8t 3 f U 7 7 A |1 64 / / 87 53 f p(z) q(z) = )3 WV p(A) = Aq(A) 7 53 64 & d(z) = p(z) − q(z) 1 )3 & 64 A 7 4 / #1 T( T' #3 7 64 7 4 #3 1 N3 54 & 4 x( / 3#3 (j7 =9=94 3T3 )4 ( ψ(z) = q(A) >= ( 53 l( p(z) q(z) 7 4 cH53 (j= 6(j( 1 )3 54 M( / 3T3 lA(j7 =j=?4 p(A)
(133) Q/ A= 3 N ? = 4 # 1 X R= 4 N= / / )3#( N( 7 53 / ( / 3#3 (97 R=9=?4 p(A) 4 6 j ( = N = # 1 N= / 4 )3 1 N3 54 V p(z) )(9( / p(z) / A R% #4 & >=?4 = )3 WV = 4 E1 =. 3T3 9 1) 3 )4 &WV 7 4 k ( / 3T3 {(97 R=9=?4 v
(134) Q/ 6(m= vr 7 )4j= ) 9 k ' f = f W(z) V47 / &k 'E N=?4 r "A; x
(135)
(136)
(137) U f : C → C / 92 / &k ' %9E= l( / 3T3 2(97 =j=?4 { ( 3 / kl 7 5 3 ' E = 497 53)&r H)4 54 r(z) f R(A) % #& . . s X. ni = degψ(z). WV( `#(jc3 k T&r ' {{X= 64j7 53 N&k ' i=1. r(j) (λi ) = f (j) (λi ),. j = 0, . . . , ni − 1,. 1' r f k N= 83 64 #1>= j F 6( f r T3 3 54 2( E%9 T( r f (A) = r(A) (j). (j). i = 1, . . . , s, r. L (j( 4j1` V r `.-k kX= 63^7 53 r #7 k ' 9 A N=9=94 |1'64j( M1)3 54 = V 3 / &r f )(9( / ( / 3T3 l(97 =j=?4 A P 63#3 ]= 6( )=j= 4 k 3 v%T&r ' @ r ' R%9T&r ' r / &k ' N=?4 r / T4 r } V]3 } N=?4 r A / &k ' 3T3} >=?4 k B = r4 r V A ( 7 ,/ + (97 4 64 k 'r7 53 ' B U A ∈ Cn×n xh 54 k ' 54 9 N' ' T3M( H / kX= & y Z −1AZ = diag (J , . . . , J ) % ' ) ! ' ! ) '
(138) ! 1 r A. . )' !. . f (z) . . Jk = . λk. 1. . . ∈ Cmk ×mk . 1 λk. ! % % ) * * ! !$ % '1)' ! % !"! % . λ(A) . f (A) = Zdiag (f (J1 ), . . . , f (Jr )) Z −1 ,. "K. %% ' .
(139)
(140) . '. . f (Jk ) = . f (λk ) f 0 (λk ) · · · f (λk ). . Jk. r(λ) = f (λk ) + f 0 (λk ) (λ − λk ) + · · · +. . f 0 (λk ) f (λk ). 0. G
(141) y t 3 7 53 { = 497 53)&k 7 64 . f (mk −1) (λk ) (mk −1)!. . . . ". 63T3%9 T&k ' @ 9N= 9$&. f (mk −1) (λk ) (λ − λk )mk −1 . (mk − 1)!. $ N3y 'k = WV # 3^7 )3 ' = 64j7 53 )&r ' ( / 3#3 l(97 R=9=?4 A N ' ' ,& f. r. f (Jk ) = r (Jk ) = f (λk ) I+f 0 (λk ) (Jk − λk I)+· · ·+. WV . / #1N3r = ". 7 4 rlH)3 (m= 6(j( 1)3 54 . f (mk −1) (λk ) (Jk − λk I)mk −1 , (mk − 1)!. 2. $ / R87 54m= X= D7 4 )7 4 =WZ63#3` / &r ' 9 N=94 r!( ' ( 4 E=9= R90( 6H / r = . 3 k & y]y [ -*. % A ∈ Cn×n % B ∈ Cn×n ' ! ' % ' % % * ! ! ' ! ! , ' % Z ∈ Cn×n % A = ZBZ −1 % % ' ) !" % % f (A) f (B) ' ! ' % f (A) = Zf (B)Z −1 G
(142) y 0 6R4 T7 )= 6( A = ZBZ −1 / Ak = ZB k Z −1 i + ( #H'E%9 V 7 4R 5H' 7 53 ' p(z) ( V9$& . p(A) = Zp(B)Z −1 .. $ )3E=?4 79N4j= ( ( = / 3#3 p(A) ( / 3T3 WV p(B) 1` k1'49( / A B 9V 83 c(j= 6(j( 7 )3 ' $ )3T3 R9% #k& f = WV4 / !&k ' >=?4 k ( VtWV6( T(m= / G7 53 {X= 64j7 53 N&k ' r(z) = N3WV & f (A) = r(A),. 7 'WV,(Q1)3r3l4 3 N&k . f (B) = r(B). p(A) = Zp(B)Z −1. 97 N4j= k 53)4 Q( V9 &. f (A) = Zf (B)Z −1 .. "J.
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(144) . . y]y u . . A ∈ Cn×n. 2. * ! % , ' ! % % ' . A = diag(A1 , A2 , . . . , Ar ). %% ' f (A) = diag(f (A ), . . . , f (A ))
(145) y U` 3#3 )T3M4 7 & 53 ' {X= 64j7 53 N&k ' H)4 f (z) ( / 3T3 (j7 =9=94 r(z) ) A 1. r. . f (A) = r(A) = diag(r(A1 ), . . . , r(Ar )). 2. . ! # . $ >= / >=?4 k. A ∈ Cn×n. / 9 N=?4 r. X. / 9 c4 k pF 6( A ( . X p = A.. ,. P 6 3.k)( x( N3 )4 n = 1 5H' / 64 k 7 3 6(j( #1'49( t9&k4 V p 4 r #(j= = C r7 64 n > 1 / 9 4 k pF 6( Q7 /,+ ' ( #(j= 4 R 7 5(j( ' (9( 64 r !9% E= Q( 53 / k& '= ,R 0 4&( k87 3 N=?4 k A=. . 0 1 0 0. . X=. . a b c d. . GV9c4 r / 4 N= ~t !N=9= ( / 7 7 ' WV 3 2H'k4 N=94 k ( / 9 4 k / !4 N= ( ( N=9= k #3 ( T(m= k . A. 3T3 54 ( 87 ' 3 k T&r ' . 2 a + bc = 0 ab + bd = 1 ca + dc = 0 cb + d2 = 0. X2 = A. WV7 4 + '0 =j= Q( 53 / &r ' s r1'49( 3 N=?4 r
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(147) !%9E= 4 k / 4 N= / X= 5H { N=94 k3T3 h )4 Y =. . ±1 0 c ∓1. "+. . .
(148)
(149) . / a( / a4 r / 4 N= $& k#3l1'4 E%9N4 !N=j= V Y = I 7 4 )H' c ∈ UC3
(150) N=?4 k '2( H' )3 )4 V9 ( k87 4 / 9 4 r pF 6( ' 7 64 / X= A 4 #H / N4 9 1'rkc3 N=94 k2( H' )3 )4 3 6( #(j= k &kt3T324 r pF 6( T7 r 9N3T3 (j=94 / =9= / 4 r= #1`T( 54 63 kk = )4 V k 54j4 #(97 ' ' NH53 / = 1 )3 54 2. A &r64 A 4 = / =9= 3 4 k pF 6( / 63T3 7/ X= 4 6(j( X= !" # $ % &. 3 4 r pF 6( . x
(151)
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(153) ]d U / 7!7 V 83 N=?4 r '{( H 53 N4 A ∈ Cn×n N' ' / = 1 )3 )4 Λ(A) = {λi | i = 1, . . . , n} r ' arg(λi) 6= π ∀i 3T3 54 3 4 k 6( !" # $ % WV WV64 k k ' A1/p ∈ Cn×n * 3 N=?4 k V p( `F T(mc 3 r ( 6H / kX= (r '#&k & . • (A1/p )p = A • arg(Λ(A1/p )) ⊆ (−π/p, π/p). t N3#=?4 r79N4 53 . A 9V / = 1 )3 54 =3 pF ( R=9= 54 & 1/p. Sp = {z ∈ C, z 6= 0 : −π/p < argz < π/p}.. 4 k ( 7 4 k#79)3 N=94 k X= 4 6(j( X= [()([< )3#=94 / = T3 3{k)3r 53 p F / &r ' = N=94 k / )3 3 * ! ' !" % % ' )= 3R=9= 4 N= / 4 r ' T 3 '
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(155)
(156)
(157) '0 / = 1N 35 4 ' 6 H'N = E 1 ( * ! ' !" % % ' A W V Ar )∈= 6C4 k r ' F R %9# = r ' & sect (A) . . . . n×n. n. 1'. sectn (A) = (An )−1/n A,. 3 2 4 r nF 6( 27 4 k#79)3 A 0 64 n = 2 3G / !&k sect (A) =j= 3G / &r ' ,! ' A 1 k kN= k ' (An )1/n. n. 2. / &r 'sign(A) sign 6(m= r #3.r 'r=9= O ( 6H " F " / / 64 tk F 7 3 6(j( )3T3 N=?4 r k R%9E= ( 53E= X= 7 4. N=94 k A ∈ Cn×n WV[ '2V9 / = 1 )3 )4 = NH' 9)4 M7 / 4 s )3TH ' l3 r( 6H / kX= 7 4 )7 4 =WZ ,Nk & H53 / = 1 )3 54 ' sect (A) ( ' 4 k nF 6( 1 / = sect (A)n = • In. '. n. n. "O.
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(159) . = / =9= H53 E / = 1N3 54 {. ( . ' ( V Z ( ' N=?4 r= G( !H' 53)4 63#3 l(m= 6(j( 54 )3T3 54 • • A sectn (A)−1 = sectn (A)−1 A 63( R=9= 54 (− π , π ) n n. Asectn (A)−1. sectn (V −1 ZV ) = V −1 sectn (Z)V,. ( sect (Z) 9% #= n t k 'r=j=
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(162)
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(173) p ! * ! ! ' ! ' λ ) p
(174) p ' ! ' * ! ' ! 2 J ' (' % ! ' ! J G
(175) y & ( / &r ' h4 k pF 6( f (λ) = λ WV N4 kX= <%#= r( / 3T3 1/p. k. k. . k. (97 R=9=94 63e' 3 `rk 54 9 J
(176) Q/ e3rh 54 / 3 ; 7 6 4z3 p 4 r9T(m= = k ( 6 H T 4 R 9 = = r . = 9)3M= r 54 r @! / fj (Jk )(j= vJkv 5(j=94 )4 O V (j= 4 k ( ' 3 / WV 4 k pF 6( B / p =?4 'H 53 N4 ( / 7 64 54 { J U / 7 7 ' V 3 k>=?4 k 1/p. . X=. x1 1 · · · 0. ,) . x 1 mk x mk mk. . ,.
(177) 6^
(178) ( / 9c4 k pF 6( =94 H' 53N4 ( / 7 4 )4 . λk. X p = Jk = . $ H / )H)3 H53 3kkX= < 7 )( #&k . Jk. 5(j( $& . 1. . 1 λk. . )(j=?4 )4 7 64R / &r ' WV 1N3TH' ' r '3& xpii = λk. xi i+1. p−1 X. c(i, 3 r4i)3 )& (i, i + 1). X p = Jk. ( 7 /$+ @. 1 ≤ i ≤ mk. K. xp−1−r xri+1 i+1 = 1 1 ≤ i ≤ mk − 1 ii. U 3 / = 1 )3 54 λ V9l3 6(97!4 6(j( ' 7 53)4 @ ( ' 9N= 9$& k r=0. . xi i = |λk |1/p ei(θ+2πk)/p. T3 3 54 H)3E3r r = EG7 5( #&k ' kX= 3 r( H / kX= h )4 &. (i, i). λk = |λk |eiθ. 3 p 4 k pF 6( . k = 0, 1, . . . , p − 1. (i + 1, i + 1). X V9 4 T(j7 =j= #1 F. xi i = |λk |1/p ei(θ+2πk1 )/p. xi+1 i+1 = |λk |1/p ei(θ+2πk2 )/p. r '. 7 /$+ k) 13T3 , 5k4 2∈4 T( {0, 64 E1'1,64 . . .r, 'p−& 1} p−1 X r=0. xri+1 i+1 xp−1−r ii. =. (G( ' WVk 79N4 3T3h 54 / 3 K- ( . p−1 X r=0. = |λk | = |λk |. |λk | p−1 p. p−1−r p. e. iθ p−1 p. ei(θ+2πk1 ) p−1 X. p−1−r p. ei2πk1. r. r. |λk | p ei(θ+2πk2 ) p. (p−1−r) p. r. ei2πk2 p. r=0. p−1 p. e. i(θ+2πk1 ) p−1 p. p−1 X. r. ei2π(k2 −k1 ) p .. $ N3T3 . / )&k ' K ( zV90WV 3 ( ' tWV r ' 7)4 63#3 6(97!4 6(j( ' 7!4 kr F r = 'G7 /$+ 6(j( 6 4 / H / N3 l&k4 / & p−1 X r=0. r=0. r. ei2π(k2 −k1 ) p 6= 0.. ,D".
(179)
(180) U 7 4 k k = k )3T3 54 ( 27 4 rkrkX1= E1'2kX= $&. xi i. p−1 X. . xi+1 i+1. )(9( / ' 3 (m= (9( 1)3 54 3 . r. ei2π(k2 −k1 ) p = p.. U 1 kk r ' ( 4 x x k 'z4 r ii i+1 i+1 E 1 ' 6 9 4 ( 9 ) H 4 ) r & ) # 3. 3 k . = E=WZ & k k r=0. 1. 1−e. λk. 4 =4 kX= )3T3 54 . 2. i2π(k2 −k1 ). . = 1−e. i2π(k2 −k1 )/p. . 1+e. i2π(k2 −k1 )/p. 3l( 27 4 rkrkX= *( 7 /$+ 4 #( 64 #1'4 r '& p−1 X. r. ei2π(k2 −k1 ) p =. + ··· + e. i2π(k2 −k1 ) p−1 p. . 1 − ei2π(k2 −k1 ) = 0. 1 − ei2π(k2 −k1 )/p. C / 6 (m= )(9( / 4 7 4 WV 3c( ' 7 /,+ 6(j( 4 / 3#3 / (kr6(9( F 4 kX= k = k U 6H / W1Vl3 2 >=?4 k X V93 )H' ')3 r 5(j= = 9)3y 'k = WV X 7 $/ + 6(9( 64 / kX= R= 64 9N= 79)4j= T4 R9 0( / ' 63 kkX= 9)H' '9N3 ;!" ( E N=9= k 3 2= 6( r=0. 2. t 3 = r 54 r K 5(j=94 V2= / =j= l3 24 ry / ' 3 `rk 5 4 9 ( ' / !&k { J r 5( 7 64R%T&r ' ( ' 87 )3 ' J t ! N=j= M1)J3 k& k. k. fj (Jk ) =. m k −1 X. (r). fj (λk )C r ,. r=0. 1'N' ' 27 5(j= . . C= . 0. 1 0. . 1 0. (4 r pF ( A WVx( ' / &k ' A ( ' 7 ) 3 ' A 7 64 R%9#k& s r1'49( = / =9= 3 4 k pF ( R / > =?4 k ( ' rk6( F ( N4 r = Q / &r {63#3 N=?4 r i ' ( 64 t6( r87 c3 >=?4 k & X=. . 1 a 0 −1. ,5,. .
(181) 6^
(182) . ' ' X = I X ' / 7 53 ' I ~$ k ' ( 6H / k &k23 N=94 k kX= E=WZ V9 4 k / 4 N= [WV< '2( ' / &r ` N=?4 r63D( k ( %9E= 7 4 krkr = k r = P ) ( k )3T3 54 *3 k=63)(9( #%9k)4 = / =9= r3 Q( 53 / &r ' 63 7 4 5' 3 k == 64 F 9N 4 3 4 k ( R / 9 > =?4 kR ( H' 53)4 ~B k 54 WV) '!' p F N = k ' f (J ), j = 1, . . . , p 3 p 4 Ar∈ pCF 6( r=?4 !H' 53)4 ( / 7 6 4 5 4 E 3' 3 `rk 5 4 9 J 1'k = r5 4 r K-R 4 # ( / 3E= N = 7 6 4Q3 / 5 (j=94 ) &r ( 4 i ' ( 4 6 (j( / 2 2. n×n. j. "; & y '
(183) !. k. . k. . . A ∈ Cn×n. * ! % ! ' ! ! , ' % ! % ' ) ! ' ! ). Z −1 AZ = diag(J1 , . . . , Jr ) = J.. . %% ' % * % % %
(184) p ' j ) ' ! J . X. A ' ! '
(185) %
(186) . . X = ZU diag(fj1 (J1 ), . . . , fjr (Jr ))U −1 Z −1 , k. ∈ {1, . . . , p}, k = 1, . . . , r. . U . * ! % % )' * % . t 3 = r 54 k V 8( 6H / 87 64 R=9= 4?Z 6( 4 E1'64 83(j=?4 / =9= / 4 63#3 N=94 r WV r '879)4 63 = r 54 r J! U y ] *
(187) (' ! ' % % A ∈ Cn×n . % ' ) ! ' ! ) '
(188) % % ! Z −1AZ = diag(J1, . . . , Jr ). % * % ' % * ' ! AX = XA ' ! '
(189)
(190) . '. X = ZW Z −1 ,. % % % ' )' ! 0 W = T * ! % ' % % *$
(191) ) % . W = [Wi j ]. . ij. Wi j ∈ Cmi ×mj . % % . λi = 6 λj λi = λ j. ij. ' T m < m % ' T = [0 U ] ) ' ! U 0 5(9( )3T3 54 / ( N 4 T 3 4 T( / 3#= >= 063z= k 54 k J 7 643 )(j( E%9)4 8= / =j= 3 4 r ( / N=?4 r '( H' 53)4 / H54 / 7!7& / 63T3 WV ( ' p 8F 7 53 ' A / 63T3 WV 'G3 l( 'A ∈ CH54 )&k N3 ( H / kX= = r 54 r ij . ij. i. j. ij. ij. n×n. y w 0||
(192) !
(193) a ww G0
(194)
(195) p |
(196) y
(197) y D>
(198) a |
(199) w A ∈ Cn×n * ! % ! ' ! ! , ' % ! % ' ('
(200) ! A = ZJZ −1 = Zdiag(J1 , J2 , . . . , Jr )Z −1 ,. ,);.
(201)
(202) . '. %% ' . . % (! * m ' + m -% * %+' . . % .' + m % = ! n% A ! p p ' ! ' * ! ' ! A
(203) % . Ji ∈ Cmi ×mi s≤r A.
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