Autovalori, autovettori e diagonalizzazione delle matrici

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DXWRYDORUL H DXWRYHWWRUL1
Vld xqd pdwulfh txdgudwd gl ruglqh ? uhdoh +r frpsohvvd, h xq yhwwruh
frorqqd1
Vh ' b frq b 5 U +r F, h 9' f/ doorud b ë ghwwr dxwrydoruh gl h ë o*dxwryhwwruh fruulvsrqghqwh d b1 Vl lqglfd frq Tb o*dxwrvsd}lr gl b/
flrë
Tb ' i G ' bj ^ ifj1
Tb lqfoxgh wxwwl jol dxwryhwwrul hg dqfkh lo yhwwruh qxoor hg ë xq
vrwwrvsd}lr gl U? +r gl F? ,1
RVVHUYD]LRQL +lpsruwdqwl$,1
4, Vh b ë xq dxwrydoruh gl frq dxwryhwwruh / doorud b6 ë xq dxwrydoruh
gl 6 frq jol vwhvvl dxwryhwwrul/ shu rjql 6 lqwhur srvlwlyr1 Lqidwwl/ shu
6 ' 2/
2 ' ' b ' b ' bb ' b2 1
Door vwhvvr prgr vl surfhgh shu 6 jhqhulfr1 Vh ë lqyhuwleloh/ or vwhvvr
ydoh shu rjql 6 lqwhur qhjdwlyr1
5, Dxwrvsd}l gl dxwrydorul glvwlqwl kdqqr lq frpxqh vror lo yhwwruh qxoor/
flrë dxwryhwwrul gl dxwrydorul glvwlqwl vrqr lqglshqghqwl1 Lqidwwl/ vh
5 Tb h 5 T> , ' b h ' > , Eb > ' f
pd b 9' > , ' f1
Frph vl wurydqr dxwrydorul h dxwryhwwrul1
' b frq 9' f vl sxõ ulvfulyhuh frph E bU ' f/ fkh ë xq vlvwhpd
olqhduh rprjhqhr frq vrox}lrql qrq edqdol 9' f vh h vror vh
ghwE bU ' f 1
4
Txhvw*xowlpd ë o*htxd}lrqh ghjol dxwrydorul1 +Sxõ dffdghuh fkh xqd r
slü vrox}lrql vldqr frpsohvvh> lq txhvwr fdvr qrq vrqr dxwrydorul vh vl frq0
vlghudqr pdwulfl uhdol fkh rshudqr vx U? 1 Lqyhfh/ wxwwh oh pdwulfl frpsohvvh
fkh rshudqr vx F? kdqqr dxwrydorul,1
Lqglfkldpr frq Eb od vhjxhqwh hvsuhvvlrqh=
Eb ' _i|E bU1
Eb ë xq srolqrplr lq b/ ghwwr srolqrplr fdudwwhulvwlfr gl c hg kd
judgr ?1 Vl sxõ yhulfduh vxelwr fkh lo frh!flhqwh gl b? ë E? / phqwuh lo
whuplqh qrwr ë _i| +lqidwwl/ lo whuplqh qrwr gl xq srolqrplr ë Ef h/ qho
fdvr gho srolqrplr fdudwwhulvwlfr/ Ef ' _i|E fU ' _i| ,1
GLDJRQDOL]]D]LRQH GHOOH PDWULFL1
Xqd pdwulfh vl glfh gldjrqdol}}deloh vh hvlvwh xqd edvh idwwd gd dxwryhw0
wrul1 Lq txhvwd edvh ë vfulwwd lq irupd gldjrqdoh frq jol dxwrydorul vxood
gldjrqdoh1
F*ë xq lpsruwdqwh whruhpd fkh glfh fkh oh pdwulfl uhdol h vlpphwulfkh/
flrë wdol fkh ' |/ vrqr gldjrqdol}}delol +ryyldphqwh qrq ë reeoljdwrulr
fkh xqd pdwulfh vld uhdoh h vlpphwulfd shu hvvhuh gldjrqdol}}deloh,1 Qho fdvr
|
|
frpsohvvr/ od frqgl}lrqh gl vlpphwuld ë ' c gryh ë irupdwd gdl
frpsohvvl frqlxjdwl ghjol hohphqwl gl |1
Shu xqd pdwulfh uhdoh 2 2 lo ulvxowdwr ë ryylr=
@ K
ë vlpphwulfd vh K ' S1 Wuryldpr jol dxwrydorul= _i|EbU ' f
S _
glyhqwd
@b
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' f / E@ b E_ b K2 ' f
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/ @_ n b2 E@ n _b K2 ' f / b2 E@ n _b K2 n @_ ' f1
{ ' E@ n _2 n eK2 e@_ ' @2 n _2 n eK2 e@_
' E@ _2 n eK2 A f1
5
Vh { ' f/ doorud K ' f h od pdwulfh ë jld gldjrqdoh1 Vh lqyhfh jol dxwrydorul
vrqr glvwlqwl iud orur +{ : f,/ jol dxwryhwwrul irupdqr xqd edvh/ shufkë vrqr 2
h vrqr lqglshqghqwl shufkë dssduwhqjrqr dg dxwrvsd}l gl dxwrydorul glvwlqwl1
Vh ë gldjrqdol}}deloh/ yxro gluh fkh fdpeldqgr edvh/ flrë sdvvdqgr dg
xqd edvh idwwd gl dxwryhwwrul/ glyhqwd ' 3 / frq gldjrqdoh1 Od
irupxod qrq ë xqlfd/ qho vhqvr fkh txdovldvl pdwulfh fkh kd frph
frorqqh oh frpsrqhqwl gl xqd edvh gl dxwryhwwrul/ wudvirupd lq
xqd pdwulfh gldjrqdoh1
FRQGL]LRQL GL GLDJRQDOL]]DELOLWÃ1
Vld _ Eb od prowhsolflwã dojheulfd ghoo*dxwrydoruh b +flrë od prowhsolf0
lwã gl b frph udglfh gho srolqrplr fdudwwhulvwlfr, h 6 Eb od prowhsolf0
lwã jhrphwulfd gl b +flrë od glphqvlrqh ghoo*dxwrvsd}lr Tb,1 Vl kd vhpsuh
9 6 Eb 9 _ Eb1 Shu fxl/ qho fdvr frpsohvvr +flrë qho fdvr gl pdwulfl
rshudqwl vx yhwwrul gl F? ,/ vlffrph xq srolqrplr gl judgr ? kd vhpsuh ?
udglfl/ vl kd vhpsuh
S
_ Eb ' ?/
S
gryh
ë od vrppd vx wxwwl jol dxwrydorul1 Doorud/ vh shu rjql b vl kd
6 Eb ' _ Eb/ flrë jol dxwrydorul vrqr wxwwl uhjrodul/ ë fkldur fkh od pdwulfh
ë gldjrqdol}}deloh/
S shufkë fl vrqr deedvwdq}d dxwryhwwrul lqglshqghqwl shu
irupduh xqd edvh
6 Eb ' ?1
Qho fdvr uhdoh/ txhvwr qrq edvwd/ shufkë
qrq ë ghwwr fkh fl vldqr ?
S
udglfl
uhdol h txlqgl qrq ë jdudqwlwr fkh
_ Eb ' ?/ txlqgl qhpphqr fkh
S
6 Eb ' ?1 Doorud/ xqd
pdwulfh
uhdoh
ë
gldjrqdol}}deloh
vh _ Eb ' 6 Eb
S
shu rjql b h/ lqrowuh/
_ Eb ' ?1
FDVL SDUWLFRODUL1
Vh xqd pdwulfh frpsohvvd kd dxwrydorul wxwwl glvwlqwl
ë gldjrqdol}}0
S
deloh1 Lqidwwl/ lq txhvwr fdvr/ _ Eb ' shu rjql b h
_ Eb ' ? h txlqgl
ghyh hvvhuh dqfkh 6 Eb ' / txlqgl vldpr qho fdvr gl dxwrydorul uhjrodul1 Xqd
pdwulfh
uhdoh fkh kd dxwrydorul wxwwl glvwlqwl ë gldjrqdol}}deloh vh/ lqrowuh/
S
_ Eb ' ?1
Xqd pdwulfh ghood irupd
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gryh c 2 c c r vrqr pdwulfl txdgudwh/ ë ghwwd pdwulfh d eorf0
fkl1 Vl sxõ glprvwuduh fkh lo vxr ghwhuplqdqwh ë xjxdoh do surgrwwr ghl
ghwhuplqdqwl ghl ydul eorffkl c 2c c r 1 Lqrowuh/ shu or vwxglr gl dxwr0
ydorul/ dxwrvsd}l h gldjrqdol}}delolwã vl srvvrqr frqvlghuduh l ydul eorffkl
vhsdudwdphqwh1
RVVHUYD]LRQH1
Vh xqd pdwulfh ë gldjrqdol}}deloh wxwwh oh vxh srwhq}h lqwhuh 6 or vrqr1
Vh ë lqyhuwleloh h gldjrqdol}}deloh/ dqfkh oh vxh srwhq}h lqwhuh qhjdwlyh or
vrqr +edvwd ulfrugduh fkh h 6 kdqqr jol vwhvvl dxwryhwwrul,1
HVHPSL1
4, Wuryduh xqd pdwulfh fkh gldjrqdol}}d
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2

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Ghwhuplqldpr jol dxwrydorul=

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Doorud
ë o*dxwrvsd}lr uhodwlyr d b h xqd edvh ë/ dg hvhpslr/ '

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Doorud o*dxwrvsd}lr T2 ë frvwlwxlwr gd yhwwrul gho wlsr

1
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Rud srvvldpr vfulyhuh / phwwhqgr lq frorqqd h 2=
'

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Eb ' _i|E bU ' _i| C b
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frq prowhsolflwã 2
frq prowhsolflwã
Dxwrvsd}l= T ' i G ' j> T2 ' i G ' j
3
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T ë 3
doorud4lo sldqr gl htxd}lrqh % + ' f/ flrë 3
% '4
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xq*htxd}lrqh h vvldpr xq sdudphwur1
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qrq ë gldjrqdol}}deloh1 Lqidwwl/ od pdwulfh dvvrfldwd ë
3
4
f f
' C f f D
f f
h lo srolqrqlr fdudwwhulvwlfr ë Eb ' E b b2 n 1 Xq vror dxwrydoruh
ë uhdoh b ' 1 Doorud T kd glphqvlrqh h xq yhwwruh qrq edvwd shu irupduh
xqd edvh lq U1 ë lqyhfh gldjrqdol}}deloh vx F/ flrë frph hqgrpruvpr
gl F / shufkë lq F kd wuh dxwrydorul glvwlqwl= b ' / b2 ' / b ' h l
fruulvsrqghqwl dxwrvsd}l kdqqr wxwwl glphqvlrqh 41
7, Od pdwulfh
3
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ë d eorffkl= '
c 2 ' C 2 2 2 D1 Lo sulpr eorffr kd
f S
2 2 2
dxwrydorul h S +od yhulfd ë lpphgldwd,1 Lo vhfrqgr eorffr kd udqjr / txlqgl
od glphqvlrqh gho qxfohr ë 2 +_4 !ih 2 ' oE2, h f ë xq dxwrydoruh gl
prowhsolflwã jhrphwulfd 2 +lqidwwl/ _4 !ih 2 ' _4 !ihE2 fU ' _Ef,1
Lqrowuh/ ulxvfldpr d wuryduh vxelwr xq dowur dxwrydoruh/ rvvhuydqgr fkh
3
3
43 4 3 4
3 4
2 2 2

S

C 2 2 2 D C D ' C S D ' S C D/
2 2 2

S

shu fxl S ë xq dxwrydoruh1 O*dxwrydoruh kd prowhsolflwã dojheulfd h jhr0
phwulfd xjxdoh d 1 Shu txdqwr uljxdugd o*dxwrydoruh S/ txhvwr kd prowhsolflwã
dojheulfd xjxdoh d 2/ phqwuh vl ghyh wuryduh od vxd prowhsolflwã jhrphwulfd=
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ghwwr vrsud1 Dqfkh od prowhsolflwã dojheulfd ë xjxdoh d 2/ shufkë od pdwulfh
ë gl ruglqh D/ shuflõ od vrppd ghooh prowhsolflwã dojheulfkh ghyh gduh D=
6Ef ' D 6ES 6E ' D 2 ' 21 Srlfkë wxwwl jol dxwrydorul vrqr
uhjrodul/ od pdwulfh ë gldjrqdol}}deloh1
WHRUHPD GL FD\OH\0KDPLOWRQ1
Vld xqd pdwulfh1 Deeldpr ylvwr fkh E% ' _i|E %U ë xq srol0
qrplr gl judgr ? qhood yduldeloh % h jol }hul gl E% vrqr jol dxwrydorul gl 1
Frqvlghuldpr lo vhjxhqwh lpsruwdqwh=
Whruhpd gl Fd|oh|0Kdplowrq1 Rjql pdwulfh ë udglfh gho vxr srol0
qrplr fdudwwhulvwlfr1
HVHPSLR1
Vld gdwd od pdwulfh
'

Srolqrplr fdudwwhulvwlfr= E% ' _i|E %U ' _i|
%

%
flrë E% ' %2 % % n ' %2 e% n 21
Lo whruhpd glfh fkh/ vh do srvwr gl % phwwldpr h do srvwr gl phwwldpr
o*lghqwlwã U/ rwwhqldpr od pdwulfh qxood/ flrë
E ' 2 e n 2U ' /
frq pdwulfh frq wxwwl f1 Lqidwwl

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Shu oh pdwulfl 2 2 ë vhpsuh
E% ' %2 %A o n _i| /
gryh A o ë od wudffld gl / flrë od vrppd ghjol hohphqwl vxood gldjrqdoh1
Frqvhjxhq}d lqwhuhvvdqwh gho whruhpd gl Fd|oh|0Kdplowrq ë lo fdofror
ghood pdwulfh lqyhuvd +vh _i| 9' f,1
Vld E% ' E? %? n n &% n &f lo srolqrplr fdudwwhulvwlfr1
+Vl ulfrugl fkh lo frh!flhqwh gl %? ë vhpsuh E? h lo whuplqh qrwr/ flrë
lo whuplqh gl srwhq}d f/ ë vhpsuh &f ' _i| ,1
E ' / E? ? n n & n _i| U ' / dE? ?3 n &?3 ?32 n n & Uo ' _i| U
/ dE? ?3 n &?3 ?32 n n & Uo ' _i| 3
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dE? ?3 n &?3 ?32 n n & Uo
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3 ' E2 n S bU ' E2 S n bU ' C f 2 D
e
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f 2
43

Autovalori, autovettori e diagonalizzazione delle matrici

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