Condition number of the matrix of transition to the normal Jordan form
We establish necessary and sufficient conditions for the well-conditioned reduction of a matrix to the Jordan normal form.
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| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4589 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510735086387200 |
|---|---|
| author | Grabovskii, O. I. Грабовський, О. И. |
| author_facet | Grabovskii, O. I. Грабовський, О. И. |
| author_sort | Grabovskii, O. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We establish necessary and sufficient conditions for the well-conditioned reduction of a matrix to the Jordan normal form. |
| first_indexed | 2026-03-24T03:01:43Z |
| format | Article |
| fulltext |
Y]J~ 512. 6
O. I. rpa6oee~.mutt (z-a 3BBiK. Ka~'a~tU,-I'Ioaia~bzalt)
CTYIIII-I~ OBYMOBJIEHOCTI MATPHIII IIEPEXO]]~ r
~O HOPMAJIbHO~ | ~KOP~
Necessary and sufficient conditions of well conditioned reduction of a maU'ix to the Jordan normal form
are established.
BCTaHOBJIeHO HeO6Xi/IHi fl/IOCTaTHi y~OBH ~o6pe o6yMo~neHoro 31~/~eHH2 MaTpHUi ~o HOpMas
~OpMH ~op/IaHa.
OaHaqeHH~. 3oeaeH~ ao ~op:~taablWi" ~opJtu .AV.opaa~a :~tamput~i J ~ 6 e ~ t o ao6-
pe o S y s w o a e ~ , agu4o ice~ye Di~cna ~yHKt~ia f j, U~O ~te cnaDae, malza, u~o Koacny
noaiSny Do J ~ampu~lo A ~ w ~ a so@azumu y 6uza,qDi SJS -1, De
II sll II s II <fs(l lA II), II" H--e iDooa ~op~a IA II =
Hcxafl A ~ 6 y ~ - a K a zourLneKcHa MaTpmla p o ~ i p y n • n. Toni ~ a r ~icI~e Ha-
cTynHr TBcp~KeHa~.
T e o p e e a . 3seDem~ Oo@e o#y~to~/tette moDi i mi.~b~.U moOi, t:o/tu ~tamput~a J e
aiazo~ta./tbHOlO.
,~oee~enn~. ]locmamnicmb. Hvxa~ [I A H = a , ~, 1 . . . . . ~m - - s~IacHi ~IHCJm
Ma'rpHI~i A ia KpaTHOCTm~H n i~os i~HO tZt . . . . . ~:m" KopHcry~o,mcb yHiraprIO~O
i rmapiawndcT~ eSK~OSOi n o p ~ a [1, C. 353] r a :ze~o~o I I Iypa npo Te, m o 6 y ~ , - a x a
zna~paTHa ~aTpHU~ yHiTapHo no~i6Ha ~o ZpHICyTHOi, 3Ha~eM0 r a z i yHiTapHi NaT-
pHIfi U, V, U V = E, mo A = UB V, he B ~ TpHKyTHa ~aTpmIa i3 J~eZCHKorpadpiano
nnopa~zosammH ~nacHa~H ~acaa~H, rlpHqOMy [[ B II = a. BCTaHOBHMO pO36HTT~I
n• ~aTp~U~Ha mXm 6aOKiSTaZH~qHHO~,moaa 1<j <--m 6aOZ Bjj
si~Irlosi/Iar B.rlaCHOMy ~.IHcYly ~]. BHKpeCYIHMO 3 MaTpHI.Ii D = B - ~ j E, Rg D = n -
- ~:], ~ j - r i pall0Z Ta CTO~HeUJ~6JIOKiB, OTpHMaBIIIH ~aTpmsjO D j. Mae~o dot D j
~0, ZSLaKH RgDJ=n-~j, aHamrrb, sci paZtXH ~aTpaUi D, mo SXOgaTby T/ j-~t
p~]lOZ 6JIOKiB D j , ~liHiflHO BHpahKaIOTbCa ,~epe3 iHmi ii p a ~ z ~ ; y mO JliHii~Hy
KOM6iSalIiIo p~ll~KH, LRO 3HaXOl~rbC~I BHIIIr ~J~OKOBOrO p~l]IKa Dj, ~[KUIO TaKi r
sxo~a-rb i3 HyJ~osHMH mocdpiuieHTa~H, ocKix~KH Dj I = O, . . . . Dj,j_I = O. Kpi~
lIboro, Dj + I,j = O, . . . . Drn j f O, ~IKI/IOLIi6JIOKI4e, OT)KC, Djj = O Ta B jj = ~,j E, ]IC
j = 1 . . . . . m, E ~ 0~HHW~Ha ~a'rpHRa ~:j-ro p o ~ i p y . PoarJ~aHeMO noc.ui~osHic'n,
~a'rpHIR,
G 1, G 2 . . . . . Gm; G p = {G:k}, (1)
mo sM3waqazrrbcs Tag: G t = B; ga~i, ~ 1 < p ~ m; G~ = Bjj za j = 1 .... , m;
Gyk =Bjk za j;k>p; pcmTa 6J~ozia--nyJ~osi; Gm=J. ~ zoamoro p, I ~ p <
m, no6yllyeMo nOCJ~OBHic~ MaTpmlb HP, H p+I, . . . , H m, he HP f GP; H q=
= {H]t} =QqHq"IP q, l~e QqPq= E; p < q ~ m ; TyT Pg, Qq--~aTpHIfi m •
6J1ozis, ZO)KHa 3 Jnmx si~pi3HaeT~Ca s i~ o~mmqHoi a tone 6JIOKOM Z isllezcaMI4 p,
q: : c . . Q,% : - c . .
�9 O, I. I"PABOBCbKH~ 1999
120 ISSN 0041.6053. YKp. ~tam. ~. ptt,, 1999 , m. 51, N e I
CTYlIIHI:, OBYMOB.rIEHOCTI MATPHHI I'IEPEXOAY ... 121
c.q = (~.q - ~..)-LVT, qk (2)
~pH TaKHM tlHHOM BH3HaqCHHX MaTpHILqX Qq P q p-lf p~IR0g Marp~Ri H q 3a~o-
BO./IbH~lr peKypCHTHe cniBBi~-10meHHa
H~ = H q - l - C p q H q - l ; (3)
yrU, oMyps~IxyAnaBcix k<_q, kv~p Ma~Mo Hg k = O; Hgp = X ~ ; a.ag Bcix jv~p
~i~CHIOCTbC~ piBHiCTb
H q = H q-j = . . . = H~ = G~' = G~ +1.
Y Hac H m = G p+ 1, OT~KC, ~RYI$ IIoc~i~OBHOCTi (1) MOT~Ha no6y~ynaTH HacTyHHr pe-
KypCHTHe cIIiBBi~HOm~HH}I:
Gp+l = LPGPKp ' L p = Q m Q m - ! ... Qp+l
Kp = p p + l p p + 2 . . , pm.
TyT 1 < p < m, L P KP = E. 3Bi~C~
J = G m = LmL m-1 . . . L 2 G ~ K 2 K 3 ... K m,
mo piBHOCR.n~HO B = G t = KJL, Ae K = K2. . . g m , L= Lm. . . L 2, K L = E, Hepe-
MHoYKylOqH, oTpHMyeMo, mo MaTpHRi K = { Kjk } Ta L = { Ljk } BH3Ha'-IalOTt~I TaK:
K~= Z (C, opCp,,...Cp,_,p, ), s<-.k-j, L j k = - C j ~ ,
j= po<p|<.., ps=k
~ e k > j ; K j j = L j j = E , j = 1 . . . . . m, g j k - - L j k = 0 RJIR Bcix k <j .
h cn iss i~aome~ (2) i (3) ~a~o~a~o oRiarm:
IIHqll _< IiHq-'ll+ll , llllH?ll <_
~e A~0=l~.~-~.~l , l<p<q<m.
OCKin~KH
I]H;II <-~, II Hq II <- ~(x + ~ :,;'.,+,) ... (x +,~:'~q),
TO
A.~a q>_p+2, Ta ~Cpq~ <- aA~lq ga.a q : p + l . Ane A:UBV=UKJLV, oTozc
y TBepAz(eHHi A =$JS -I MO~eMO noKnac'm 8 = UK, S-I = LV. OCmabKH U,
V--yHiTap.i~aTp.Ui, TO IISII=IIKII +a IIS -I II=IILII, Ae
UKjkU <- Mj, = ~ NpopiNp, l~...Np._,p, s<k-j;
jfpo<Pl <...< ps=k
ISSN 0041-6053, Yr, p. ~tam. ~. ptt.,1999, m. S l , N e I
122 O.I. FPABOBCbKPI~t
Ilgnll- IILj~II= ~-j, j " 1 . . . . . m ;
II KjkII = II L~kll = 0,, j > k.
OT~Kr rlOK.rla~OMO
= , , + . , , j. =l . . . . .
HeoSxiSnicmb. HprmycTm~O, mo ~ t~ Aem~oi MaTpHrfi J, ~xa He MaC g ia rona~-
s 0 r o B l t r . r l ~ ( M i c r a ~ xo'~a 5 o a a y r,.ni'mm: YKopgana J , po3s ipy n > 1) icHye He
cna~Ha dpyHm~ia f1" TyT A o c r a ~ s o poara~lIty'rH BHrlal~OK, KO.r~I J cx.rlalIacTl~Ca
ziamz.t~ 3 oAttiei Tazoi icaiTtirm J = Jn i3 n.uacrmtd ,mCXtOM ~, = 0. Hcxatt A (E) - J
MaTpHU~, caeMerrm aZOi a 12 = ~, 0 < [ 8 [ ~ 1; pcurra emeMerrrin 3a.va4maeTbC#l TaKa.
aX y MavpHm J. Poss'~13yloqI, I pisH~tltH~ A = S J S -1 BiArIOCHO S, OTpHMyCMo S =
= diag (eS, 8 . . . . . 8 ) , 8 ;e 0. Toni S - I = d i a g ( e - 1 8 - l , 8 - I . . . . . 8 - l ) i ~ ~ c ~ a
o6yMonJieHoc'ri q MaTpm.~i A (e) oTpm~aeMo
q(A( r = 11 sll II s -~ II = -~1~12 + n - 1 ~ ' 2 2 + n - 1 ,
/le n > I; q(A(~) ) . - - ~ r ip . lel-~0, mo cynepomrr~ yMosi
q(A(~)) < f~( l la(~) l l ) - / ~ ( l l A ( ] ) l l ) = coast.
1. XopHP,,~oucox q. MaTpMUHUtt a n a t , . - M . : M.~p, 1989.-655 c.
OAep3Kalio 06.09.94.,
nicaa Aoonpat~Batms - - 23.10.97
ISSN O041-6053. Ytcp. ~zm. ~cyim.,1999. m. 51, Ne 1
|
| id | umjimathkievua-article-4589 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:01:43Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/42/8a9e3e0214d9f2f9c68d70fcab4fcf42.pdf |
| spelling | umjimathkievua-article-45892020-03-18T21:09:14Z Condition number of the matrix of transition to the normal Jordan form Ступінь обумовленості матриці переходу до нормальної Форми Жордана Grabovskii, O. I. Грабовський, О. И. We establish necessary and sufficient conditions for the well-conditioned reduction of a matrix to the Jordan normal form. Встановлено необхідні й достатні умови добре обумовленого зведення матриці до нормальної форми Жордана. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4589 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 120–122 Український математичний журнал; Том 51 № 1 (1999); 120–122 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4589/5882 https://umj.imath.kiev.ua/index.php/umj/article/view/4589/5883 Copyright (c) 1999 Grabovskii O. I. |
| spellingShingle | Grabovskii, O. I. Грабовський, О. И. Condition number of the matrix of transition to the normal Jordan form |
| title | Condition number of the matrix of transition to the normal Jordan form |
| title_alt | Ступінь обумовленості матриці переходу до нормальної Форми Жордана |
| title_full | Condition number of the matrix of transition to the normal Jordan form |
| title_fullStr | Condition number of the matrix of transition to the normal Jordan form |
| title_full_unstemmed | Condition number of the matrix of transition to the normal Jordan form |
| title_short | Condition number of the matrix of transition to the normal Jordan form |
| title_sort | condition number of the matrix of transition to the normal jordan form |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4589 |
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