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# Introdução à Álgebra Linear.pdf

DisciplinaÁlgebra Linear I19.358 materiais281.586 seguidores
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Aµj = Aµj .
Seja \u2dcBj = {v1 + w1, v1 \u2212 w1, . . . , vdj + wdj , vdj \u2212 wdj} a base de
Ker(pj(T)rj) obtida a partir das bases Bj e Bj.
Como
(T(vi), T(wi)) = T\u302(vi, wi) = (aj + ibj)(vi, wi) + (vi+1, wi+1) ,
ou
(T(vi), T(wi)) = T\u302(vi, wi) = (aj + ibj)(vi, wi) ,
temos que\uf8f1\uf8f2\uf8f3T(vi) = ajvi \u2212 bjwi + vi+1T(wi) = bjvi + ajwi +wi+1 ou
\uf8f1\uf8f2\uf8f3T(vi) = ajvi \u2212 bjwiT(wi) = bjvi + ajwi .
Logo,
T(vi +wi) = aj(vi +wi) + bj(vi \u2212wi) + (vi+1 +wi+1)
T(vi \u2212wi) = \u2212bj(vi +wi) + aj(vi \u2212wi) + (vi+1 \u2212wi+1) ,
ou
T(vi +wi) = aj(vi +wi) + bj(vi \u2212wi)
T(vi \u2212wi) = \u2212bj(vi +wi) + aj(vi \u2212wi) .
Enta\u2dco, se {(vi1 , wi1), . . . , (vik, wik)} determinam um bloco elementar
de Jordan de Aµj de ordem k,
{(vi1 +wi1), (vi1 \u2212wi1), . . . , (vik +wik), (vk1 \u2212wik)} ,
determinam um bloco de ordem 2k na forma:
J. Delgado - K. Frensel 254 Instituto de Matema´tica - UFF
Forma Cano\u2c6nica de Jordan Real
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
aj \u2212bj
bj aj
1 0 aj \u2212bj
0 1 bj aj
1 0 aj \u2212bj
0 1 bj aj
.
.
.
.
.
.
.
.
.
.
.
.
1 0 aj \u2212bj
0 1 bj aj
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
Sejam Ui = Ker(pi(T)ri), i = 1, . . . , k e Wi = Ker((T \u2212 \u3bbiI)si),
i = 1, . . . , . Sejam B\u2dci a base de Ui constru\u131´da acima e B\u2dci+k uma base de
Wi tal que [T |Wi] gBi+k esta´ na forma cano\u2c6nica de Jordan.
Como V = U1 \u2295 . . .\u2295Uk \u2295W1 \u2295 . . .\u2295W, temos que
B\u2dc = B\u2dc1 \u222a . . . \u222a B\u2dck \u222a B\u2dck+1 \u222a . . . \u222a B\u2dck+
e´ uma base de V tal que
[T ] eB =
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
A1
.
.
.
Ak
B1
.
.
.
B
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
n×n
onde:
(I)
\u2022 Aj =
\uf8eb\uf8ec\uf8edR
j
1
.
.
.
Rjnj
\uf8f6\uf8f7\uf8f8
2dj×2dj
;
\u2022 Rji =
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
aj \u2212bj
bj aj
1 0 aj \u2212bj
0 1 bj aj
.
.
.
.
.
.
.
.
.
.
.
.
1 0 aj \u2212bj
0 1 bj aj
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
2ki×2ki
, i = 1, . . . , nj ;
J. Delgado - K. Frensel 255 Instituto de Matema´tica - UFF
Forma Cano\u2c6nica de Jordan Real
\u2022 2nj = dim(Ker(pj(T))) = 2dim(Ker(T\u302 \u2212 µjI)) ;
\u2022 ki \u2265 ki+1, i = 1, . . . , nj \u2212 1 ;
\u2022 2k1 = 2rj , ou seja, k1 = rj ;
\u2022 \u3bdk(µj) = \u2212\u3b4k\u22121(µj)+2\u3b4k(µj)\u2212\u3b4k+1(µj), sendo \u3bdk(µj) o nu´mero de blocos
de tamanho 2k × 2k associado ao autovalor µj = aj + ibj, bj > 0, e
2\u3b4k(µj) = dim(Ker(pj(T)k)) = 2dim(Ker(T\u302 \u2212 µjI)k).
(II)
\u2022 Bi =
\uf8eb\uf8edJi1 . .
.
Jimi
\uf8f6\uf8f8
qi×qi
;
\u2022 Jij =
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
\u3bbi
1 \u3bbi
1
.
.
.
.
.
.
.
.
.
1 \u3bbi
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
fj×fj
, j = 1, . . . ,mi ;
\u2022 qi = multiplicidade de \u3bbi como raiz do polino\u2c6mio caracter\u131´stico de T ;
\u2022 mi = dim(Ker(T \u2212 \u3bbiI)) ;
\u2022 fj \u2265 fj+1, j = 1, . . . ,mi \u2212 1 ;
\u2022 f1 = s1 (multiplicidade de \u3bbi como raiz do polino\u2c6mio minimal de T );
\u2022 \u3bdk(\u3bbi) = \u2212\u3b4k\u22121(\u3bbi)+2\u3b4k(\u3bbi)+\u3b4k+1(\u3bbi), sendo \u3bdk(\u3bbi) o nu´mero de blocos
de tamanho k×k associado ao autovalor \u3bbi e \u3b4k(\u3bbi) = dim(Ker(T \u2212\u3bbiI)k) .
A matriz
[T ] eB =
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
A1
.
.
.
Ak
B1
.
.
.
B
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
n×n
e´ a forma cano\u2c6nica de Jordan real do operador T .
J. Delgado - K. Frensel 256 Instituto de Matema´tica - UFF
Forma Cano\u2c6nica de Jordan Real
Observac¸a\u2dco 4.4
Se as ra\u131´zes do polino\u2c6mio caracter\u131´stico de um operador T sobre um
espac¸o vetorial real sa\u2dco todas reais, a forma cano\u2c6nica de Jordan real de
T e´ igual a sua forma cano\u2c6nica de Jordan.
Em particular, se T e´ nilpotente, sua forma cano\u2c6nica de Jordan real e´
igual a sua forma cano\u2c6nica de Jordan, que, por sua vez, e´ igual a sua
forma racional.
Unicidade: Seja B\u2dc \u2032 uma base de V tal que [T ] eB \u2032 esta´ na forma
cano\u2c6nica de Jordan real. Enta\u2dco, [T ] eB e [T ] eB \u2032 podem diferir apenas pela
ordem em que aparecem os blocos associados a uma raiz do polino\u2c6mio
caracter\u131´stico de T .
De fato, seja B\u2dc \u2032 uma base de V tal que
[T ] eB \u2032 =
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
C1
.
.
.
Cs
D1
.
.
.
Dq
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
n×n
,
onde:
\u2022 Ci =
\uf8eb\uf8edCi1 . .
.
Cipi
\uf8f6\uf8f8
ei×ei
;
\u2022 Cij =
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
\u3b1i \u2212\u3b2i
\u3b2i \u3b1i
1 0 \u3b1i \u2212\u3b2i
0 1 \u3b2i \u3b1i
.
.
.
.
.
.
.
.
.
.
.
.
1 0 \u3b1i \u2212\u3b2i
0 1 \u3b2i \u3b1i
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
2hij×2hij
;
\u2022 hij \u2265 hij+1, j = 1, . . . , pi \u2212 1 ;
\u2022 Di =
\uf8eb\uf8edDi1 . .
.
Diti
\uf8f6\uf8f8
ui×ui
;
J. Delgado - K. Frensel 257 Instituto de Matema´tica - UFF
Forma Cano\u2c6nica de Jordan Real
\u2022 Dij =
\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed
\u3b4i
1 \u3b4i
1
.
.
.
.
.
.
.
.
.
1 \u3b4i
\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8
ij×ij
;
\u2022 ij \u2265 ij+1, j = 1, . . . , ti \u2212 1 .
Como [T ] eB \u2032 esta´ na forma de blocos, temos que
pc = det(xI\u2212 [T ] eB \u2032)
= ((x\u2212 c1)
2 + d21)
e1 . . . ((x\u2212 cs)
2 + d2s)
es(x\u2212 \u3b41)
u1 . . . (x\u2212 \u3b4q)
uq ,
ou seja, \u3b11 + i\u3b21, \u3b11 \u2212 i\u3b21, . . . , \u3b1s + i\u3b2s, \u3b1s \u2212 i\u3b2s sa\u2dco as ra\u131´zes complexas
distintas de pc e \u3b41, . . . , \u3b4q sa\u2dco as ra\u131´zes reais distintas de pc.
Logo: k = s ;  = q ; {\u3b11+ i\u3b21, . . . , \u3b1s+ i\u3b2s} = {a1+ ib1, . . . , ak+ ibk}
e {\u3b41, . . . , \u3b4q} = {\u3bb1, . . . , \u3bbq} . Ale´m disso,
ei = dj , se \u3b1i + i\u3b2i = aj + ibj e ui = qj , se \u3b4i = \u3bbj .
\u2022 Seja B\u2dc \u2032 = {v\u2dc1, w\u2dc1, . . . , v\u2dcm, w\u2dcm, u\u2dc1, . . . , u\u2dcp}, onde m = d1 + . . . + dk e
p = q1 + . . .+ q.
Vamos provar que
B \u2032 = {(v\u2dc1, w\u2dc1), (v\u2dc1,\u2212w\u2dc1), . . . , (v\u2dcm, w\u2dcm), (v\u2dcm,\u2212w\u2dcm), (u\u2dc1,0), . . . , (u\u2dcp,0)}
e´ uma base de V\u302 .
Como dim(V\u302) = dim(V) = 2m+ p, basta provar que B \u2032 e´ LI.
De fato, seja
(\u3bb11 + i\u3b4
1
1)(v\u2dc1, w\u2dc1) + (\u3bb
2
1 + i\u3b4
2
1)(v\u2dc1,\u2212w\u2dc1) + . . .+ (\u3bb
1
m + i\u3b4
1
m)(v\u2dcm, w\u2dcm)
+(\u3bb2m + i\u3b4
2
m)(v\u2dcm,\u2212w\u2dcm) + \u3bb
3
1(u\u2dc1,0) + . . .+ +\u3bb3p(u\u2dcp,0) = (0,0) ,
onde \u3bb1i , \u3bb2j , \u3bb3k, \u3b41i , \u3b42i \u2208 R, i, j = 1, . . . ,m, k = 1, . . . , p.
Enta\u2dco,
(\u3bb11v\u2dc1 \u2212 \u3b4
1
1w\u2dc1, \u3b4
1
1v\u2dc1 + \u3bb
1
1w\u2dc1) + (\u3bb
2
1v\u2dc1 + \u3b4
2
1w\u2dc1, \u3b4
2
1v\u2dc1 \u2212 \u3bb
2
1w\u2dc1) + . . .
+(\u3bb1mv\u2dcm \u2212 \u3b4
1
mw\u2dcm, \u3b4
1
mv\u2dcm + \u3bb
1
mw\u2dcm) + (\u3bb
2
mv\u2dcm + \u3b4
2
mw\u2dcm, \u3b4
2
mv\u2dcm \u2212 \u3bb
2
mw\u2dcm)
+(\u3bb31u\u2dc1,0) + . . .+ (\u3bb3pu\u2dcp,0) = (0,0)
=\u21d2 (\u3bb11 + \u3bb21)v\u2dc1 + (\u2212\u3b411 + \u3b421)w\u2dc1 + . . .+ (\u3bb1m + \u3bb2m)v\u2dcm + (\u2212\u3b41m + \u3b42m)w\u2dcm
+\u3bb31u\u2dc1 + . . .+ \u3bb
3
pu\u2dcp = 0
J. Delgado - K. Frensel 258 Instituto de Matema´tica - UFF
Forma Cano\u2c6nica de Jordan Real
e
(\u3b411 + \u3b4
2
1)v\u2dc1 + (\u3bb
1
1 \u2212 \u3bb
2
1)w\u2dc1 + . . .+ (\u3b4
1
m + \u3b4
2
m)v\u2dcm + (\u3bb
1
m \u2212 \u3bb
2
m)w\u2dcm = 0 .
Como {v\u2dc1, w\u2dc1, . . . , v\u2dcm, w\u2dcm, u\u2dc1, . . . , u\u2dcp} e´ uma base de V , temos que
\u3bb1j + \u3bb
2
j = \u3bb
1
j \u2212 \u3bb
2
j = 0 ,
\u3b41j + \u3b4
2
j = \u2212\u3b4
1
j + \u3b4
2
j = 0 ,
j = 1, . . . ,m ,
e \u3bb3i = 0, i = 1, . . . , p .
Logo, \u3bb1j = \u3bb2j = \u3b41j = \u3b42j = 0 , j = 1, . . . ,m , e \u3bb3i = 0, i = 1, . . . , p .
Observe, tambe´m, que se{
T(v\u2dci) = ajv\u2dci + bjw\u2dci + v\u2dci+1
T(w\u2dci) = \u2212bjv\u2dci + ajw\u2dci + w\u2dci+1 ,
ou {
T(v\u2dci) = ajv\u2dci + bjw\u2dci
T(w\u2dci) = \u2212bjv\u2dci + ajw\u2dci ,
enta\u2dco:
T\u302(v\u2dci, w\u2dci) = (T(v\u2dci), T(w\u2dci))
= (ajv\u2dci + bjw\u2dci + v\u2dci+1,\u2212bjv\u2dci + ajw\u2dci + w\u2dci+1)
= (aj \u2212 ibj)(v\u2dci, w\u2dci) + (v\u2dci+1, w\u2dci+1)
= µj(v\u2dci, w\u2dci) + (v\u2dci+1, w\u2dci+1) ,
e
T\u302(v\u2dci,\u2212w\u2dci) = (T(v\u2dci),\u2212T(w\u2dci))
= (ajv\u2dci + bjw\u2dci + v\u2dci+1, bjv\u2dci \u2212 ajw\u2dci \u2212 w\u2dci+1)
= (aj + ibj)(v\u2dci,\u2212w\u2dci) + (v\u2dci+1,\u2212w\u2dci+1)
= µj(v\u2dci,\u2212w\u2dci) + (v\u2dci+1,\u2212w\u2dci+1) ,
ou
T\u302(v\u2dci, w\u2dci) = (T(v\u2dci), T(w\u2dci))
= (ajv\u2dci + bjw\u2dci,\u2212bjv\u2dci + ajw\u2dci)
= (aj \u2212 ibj)(v\u2dci, w\u2dci)
= µj(v\u2dci, w\u2dci) ,
e
T\u302(v\u2dci,\u2212w\u2dci) = (T(v\u2dci),\u2212T(w\u2dci))
= (ajv\u2dci + bjw\u2dci, bjv\u2dci \u2212 ajw\u2dci)
= (aj + ibj)(v\u2dci,\u2212w\u2dci)
= µj(v\u2dci,\u2212w\u2dci) ,
J. Delgado - K. Frensel 259 Instituto de Matema´tica - UFF
Forma Cano\u2c6nica de Jordan Real
Reordenando a base B \u2032 de V\u302 , obtemos a base
B \u2032\u2032 = {(v\u2dc1, w\u2dc1), . . . , (v\u2dcm, w\u2dcm), (v\u2dc1,\u2212w\u2dc1), . . . , (v\u2dcm,\u2212w\u2dcm), (u\u2dc1,0), . . . , (u\u2dcp,0)}
de V\u302 tal que [T\u302 ]B \u2032\u2032 esta´ na forma cano\u2c6nica de Jordan. Como no caso
Exemplo 4.1
Seja T o operador linear sobre R3 que e´ representado em relac¸a\u2dco a base
cano\u2c6nica pela matriz
A =
\uf8eb\uf8ec\uf8ec\uf8ed
0 0 0 \u22128
1 0 0 16
0 1 0 \u221214
0 0 1 6
\uf8f6\uf8f7\uf8f7\uf8f8 .
Logo,
pc = det(xI\u2212A) = det
\uf8eb\uf8ec\uf8ec\uf8ed
x 0 0 8
\u22121 x 0 \u221216
0 \u22121 x 14
0 0 \u22121 x\u2212 6
\uf8f6\uf8f7\uf8f7\uf8f8
= xdet
\uf8eb\uf8ed x 0 \u221216\u22121 x 14
0 \u22121 x\u2212 6
\uf8f6\uf8f8+ det
\uf8eb\uf8ed 0 0 8\u22121 x 14
0 \u22121 x\u2212 6
\uf8f6\uf8f8
= x2 det
(
x`