Localization of eigenvalues of polynomial matrices
We consider the problem of localization of eigenvalues of polynomial matrices. We propose sufficient conditions for the spectrum of a regular matrix polynomial to belong to a broad class of domains bounded by algebraic curves. These conditions generalize the known method for the localization of the...
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508937765257216 |
|---|---|
| author | Mazko, A. G. Мазко, А. Г. Мазко, А. Г. |
| author_facet | Mazko, A. G. Мазко, А. Г. Мазко, А. Г. |
| author_sort | Mazko, A. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:46Z |
| description | We consider the problem of localization of eigenvalues of polynomial matrices. We propose sufficient conditions for the spectrum of a regular matrix polynomial to belong to a broad class of domains bounded by algebraic curves. These conditions generalize the known method for the localization of the spectrum of polynomial matrices based on the solution of linear matrix inequalities. We also develop a method for the localization of eigenvalues of a parametric family of matrix polynomials in the form of a system of linear matrix inequalities. |
| first_indexed | 2026-03-24T02:33:09Z |
| format | Article |
| fulltext |
UDK 512.643
A. H. Mazko (Yn-t matematyky NAN Ukrayn¥, Kyev)
LOKALYZACYQ SOBSTVENNÁX
ZNAÇENYJ POLYNOMYAL|NÁX MATRYC
We consider the problem of localization of eigenvalues of polynomial matrices. Sufficient conditions
are suggested for belonging of the spectrum of a regular matrix polynomial to a wide class of domains
bounded by algebraic curves. These conditions generalize the known method of localization of the
spectrum of polynomial matrices consisting in the solution of linear matrix inequalities. We also
develop the method of localization of eigenvalues of a parameter family of matrix polynomials in the
form of a system of linear matrix inequalities.
Robotu prysvqçeno zadaçi lokalizaci] vlasnyx znaçen\ polinomial\nyx matryc\. Zaproponovano
dostatni umovy naleΩnosti spektra rehulqrnoho matryçnoho polinoma ßyrokomu klasu oblas-
tej, obmeΩenyx alhebra]çnymy kryvymy. Ci umovy uzahal\nggt\ vidomyj metod lokalizaci]
spektra polinomial\nyx matryc\, wo zvodyt\sq do rozv’qzannq linijnyx matryçnyx nerivnostej.
Rozvyva[t\sq metod lokalizaci] vlasnyx znaçen\ parametryçno] sim’] matryçnyx polinomiv u vyh-
lqdi systemy linijnyx matryçnyx nerivnostej.
1. Vvedenye. Mnohye teoretyçeskye y prykladn¥e zadaçy svqzan¥ s analyzom
spektral\n¥x svojstv matryçn¥x polynomov y funkcyj. Spektr σ( )F mat-
ryçnoho polynoma
F( )λ = A A0 1+ λ + … + λs
sA , Ai
n n∈ ×C , i = 0, … , s, (1.1)
pry uslovyy rehulqrnosty det ( )F λ /≡ 0 , λ ∈C , sostavlqgt vse eho sobstven-
n¥e znaçenyq, qvlqgwyesq kornqmy xarakterystyçeskoho uravnenyq
det ( )F λ = 0 s uçetom kratnostej. KaΩdomu sobstvennomu znaçenyg λ σ∈ ( )F
sootvetstvugt lev¥e y prav¥e sobstvenn¥e vektor¥ u∗ ≠ 0 , v ≠ 0 , opredelqe-
m¥e sootnoßenyqmy u F∗ =( )λ 0 y F( )λ v = 0 .
Sobstvenn¥e znaçenyq y sobstvenn¥e vektor¥ matryçn¥x polynomov ys-
pol\zugtsq, v çastnosty, v teoryy ustojçyvosty y stabylyzacyy dynamyçeskyx
system pry opysanyy osnovn¥x dynamyçeskyx xarakterystyk. Esly nepo-
sredstvennoe v¥çyslenye sobstvenn¥x znaçenyj matryçnoho polynoma (1.1) ne
pryvodyt k Ωelaem¥m rezul\tatam, osobenno v sluçaqx v¥sokyx porqdkov s y
razmerov n × n, to stanovqtsq aktual\n¥my zadaçy ocenky y lokalyzacyy to-
çek σ( )F otnosytel\no zadann¥x oblastej kompleksnoj ploskosty. ∏tym za-
daçam posvqwen¥ mnohoçyslenn¥e rabot¥ (sm., naprymer, [1 – 5] ).
V [6], v çastnosty, poluçen¥ uslovyq raspoloΩenyq vsex sobstvenn¥x znaçe-
nyj matryçnoho polynoma (1.1) v oblastqx vyda
Λ1 = λ γ γ λ γ λ γ λλ∈ + + + >{ }C : 00 01 10 11 0 ,
hde γ 00 , γ 01 = γ10 , γ11 — zadann¥e koπffycyent¥. Dann¥e uslovyq ymegt
vyd lynejn¥x matryçn¥x neravenstv
AB BA C C∗ ∗+ +
γ γ
γ γ
00 01
10 11
X X
X X
T > 0, X = X∗ > 0, (1.2)
hde
A =
A
As
0
� , B =
B
Bs
0
� , C =
I
I
ns
n ns
n ns
ns0
0
,
,
.
© A. H. MAZKO, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8 1063
1064 A. H. MAZKO
Matryc¥ X y B v (1.2) podleΩat opredelenyg. V komp\gternoj systeme
MATLAB ymeetsq πffektyvnaq procedura dlq reßenyq system lynejn¥x mat-
ryçn¥x neravenstv obweho vyda.
Oçevydno, çto kaΩdaq oblast\ Λ1 pry opredelenn¥x ohranyçenyqx na
koπffycyent¥ γ ij ohranyçena nekotoroj prqmoj yly okruΩnost\g ϕ( , )x y =
= 0. V πtom lehko ubedyt\sq, polahaq λ = x + i y y ysklgçaq sluçay Λ1 = C y
Λ1 = ∅ .
Dannaq rabota posvqwena zadaçam lokalyzacyy sobstvenn¥x znaçenyj mat-
ryçn¥x polynomov y yx semejstv. V p. 2 obobwagtsq sootnoßenyq (1.2) y usta-
navlyvagtsq dostatoçn¥e uslovyq prynadleΩnosty spektra matryçnoho poly-
noma ßyrokomu klassu oblastej v kompleksnoj ploskosty, ohranyçenn¥x al-
hebrayçeskymy kryv¥my:
Λk = λ λ λ γ λ λ∈ = >
=
∑C : ( , ) :
,
f ij
i j
i j
k
0
0
, (1.3)
hde k ≥ 1 , γ ij — skalqrn¥e koπffycyent¥, sostavlqgwye πrmytovu matrycu
Γ razmera (k + 1) × (k + 1). V çastnosty, pry k ≥ s spravedlyvo sledugwee
utverΩdenye.
Teorema 1.1. Pust\ suwestvugt n × n-matryc¥ B0 , … , Bk , H H= ∗
y
X X= ∗ ≥ 0 takye, çto
AB BA A A∗ ∗ ∗+ + + ⊗H XΓ > 0, (1.4)
hde
A =
A
Ak
0
� , As+1 = … = Ak n n: ,= 0 , B =
B
Bk
0
� .
Tohda vse sobstvenn¥e znaçenyq matryçnoho polynoma (1.1) raspoloΩen¥ v ob-
lasty (1.3).
V p.C3 rassmatryvagtsq prymer¥ yspol\zovanyq poluçenn¥x rezul\tatov
dlq lynejn¥x y kvadratyçn¥x matryçn¥x puçkov, a takΩe v sluçae skalqrnoho
polynoma proyzvol\noj stepeny. V p.C4 razvyvaetsq metod lokalyzacyy sobst-
venn¥x znaçenyj semejstv matryçn¥x polynomov, voznykagwyx v zadaçax o ro-
bastnoj ustojçyvosty. Dann¥j metod svodytsq k reßenyg system lynejn¥x
matryçn¥x neravenstv. Pryvodytsq yllgstratyvn¥j prymer.
V rabote yspol\zugtsq sledugwye oboznaçenyq: ⊗ , ∗ y T — operacyy
sootvetstvenno kronekerovoho proyzvedenyq, kompleksnoho soprqΩenyq y
transponyrovanyq matryc; In — edynyçnaq matryca razmera n × n; 0n m, — nu-
levaq matryca razmera n × m; X > 0 (X ≥ 0) — poloΩytel\no (neotrycatel\no)
opredelennaq matryca X; i (X) = i X i X i X+ −{ }( ), ( ), ( )0 — ynercyq πrmytovoj mat-
ryc¥ X X= ∗
, sostoqwaq yz kolyçestv ee poloΩytel\n¥x i X+( )( ) , otryca-
tel\n¥x i X−( )( ) y nulev¥x i X0( )( ) sobstvenn¥x znaçenyj s uçetom kratnostej;
λmax( )X λmin( )X( ) — maksymal\noe (mynymal\noe) sobstvennoe znaçenye πrmy-
tovoj matryc¥, σ( )F — spektr matryçnoho polynoma F( )λ .
2. Lokalyzacyq spektra alhebrayçeskymy kryv¥my. Rassmotrym rehu-
lqrn¥j matryçn¥j polynom (1.1) y alhebrayçeskug oblast\ Λk vyda (1.3) v
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
LOKALYZACYQ SOBSTVENNÁX ZNAÇENYJ POLYNOMYAL|NÁX MATRYC 1065
kompleksnoj ploskosty C. Budem predpolahat\, çto Λk ne sovpadaet s C y
∅ . Dlq πtoho neobxodymo, çtob¥ matryca Γ ne b¥la neotrycatel\no yly ne-
poloΩytel\no opredelennoj, t.Ce. i+ ≠( )Γ 0 y i− ≠( )Γ 0 .
Sformulyruem uslovyq prynadleΩnosty vsex toçek σ( )F oblasty Λk s
pomow\g bolee obweho, çem (1.2), matryçnoho neravenstva.
Teorema 2.1. Pust\ suwestvugt matryc¥
B =
B
Bm
0
� , H H= ∗ , X
X X
X X
r
r rr
=
00 0
0
�
� � �
�
,
v kotor¥x vse Bi , H , Xij razmera n × n, m = max ( , )s k , r = m – k , y dlq
kotor¥x v¥polnqgtsq dva uslovyq:
1) matryçnoe neravenstvo
AB BA A A∗ ∗ ∗+ + +H L X( ) > 0, (2.1)
hde
A =
A
Am
0
� , As+1 = … = Am n n: ,= 0 , L X( ) = γ ij i j
T
i j
k
C XC
, =
∑
0
,
C E Ii
i
n= ⊗∆ , ∆ =
0 0
0
1
1
,
,
m
m mI
, E
Ir
k r
=
+
+
1
10 ,
;
2) vklgçenye
C\Λk Õ Ω : = λ λ λ∈ ≥
=
∑C :
,
i j
ij
i j
r
X 0
0
. (2.2)
Tohda vse sobstvenn¥e znaçenyq matryçnoho polynoma (1.1) raspoloΩen¥ v
oblasty Λk .
Dokazatel\stvo. Zapyßem operator L X( ) v (2.1) v vyde
L X( ) = C C( )Γ ⊗ X T , C = …[ ]C Ck0, , = R In⊗ ,
hde matryca R razmera (m + 1) × (k + 1) (r + 1) ymeet sledugwug strukturu:
R = E E Ek, , ,∆ ∆… =
I
I
r
r
k r
r
r
k r
r+
+
− +
+
+
− +
+
…
1
1 1
1 1
1 1
1
1 1
1 1
0
0
0
0
0
,
,
,
,
,
00 1 1
1
k r
rI
− +
+
, .
Oçevydno, çto F( )λ = Zm ( )λ A y f ( , )λ λ = z zk k( ) ( )λ λΓ ∗
, hde
Zm ( )λ = I I In n
m
n, , ,λ λ… = z Im n( )λ ⊗ , zm ( )λ = 1, , ,λ λ…
m .
Yspol\zuq strukturu matryc¥ C y svojstva kronekerovoho proyzvedenyq,
ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
1066 A. H. MAZKO
Zm ( )λ C = z I R Im n n( ) ( )λ ⊗( ) ⊗ = z R Im n( )λ ⊗ =
= z z z Ir r
k
r n( ), ( ), , ( )λ λ λ λ λ… ⊗ = z z Ik r n( ) ( )λ λ⊗ ⊗ = z Zk r( ) ( )λ λ⊗ .
UmnoΩyv matryçnoe neravenstvo (2.1) sleva (sprava) na matrycu polnoho ranha
Zm ( )λ Zm
∗( )( )λ , poluçym
F G( ) ( )λ λ∗ + G F( ) ( )λ λ∗ + F HF( ) ( )λ λ∗ +
+ z Z X z Zk r k r( ) ( ) ( ) ( ) ( )λ λ λ λ⊗[ ] ⊗ ⊗
∗ ∗Γ = F G( ) ( )λ λ∗ +
+ G F( ) ( )λ λ∗ + F HF( ) ( )λ λ∗ + f Z X Zr r( , ) ( ) ( )λ λ λ λ∗ > 0,
hde G( )λ = B0 + λB1 + … + λm
mB — nekotor¥j matryçn¥j polynom.
Pust\ λ = λ0 ∈ σ( )F — proyzvol\noe sobstvennoe znaçenye matryçnoho po-
lynoma F( )λ , y predpoloΩym, çto λ0 ∉Λk . Tohda sohlasno (2.2) f ( , )λ λ0 0 ×
× Zr ( )λ0 C X Zr
∗( )λ0 ≤ 0, y, sledovatel\no,
F G( ) ( )λ λ0 0
∗ + G F( ) ( )λ λ0 0
∗ + F HF( ) ( )λ λ0 0
∗ > 0.
Odnako, πto ne tak, poskol\ku
v0 0 0 0 0 0 0
∗ ∗ ∗ ∗+ + F G G F F HF( ) ( ) ( ) ( ) ( ) ( )λ λ λ λ λ λ vv0 = 0,
hde v0 0∗ ≠ — lev¥j sobstvenn¥j vektor matryçnoho polynoma F( )λ , soot-
vetstvugwyj sobstvennomu znaçenyg λ σ0 ∈ ( )F . Yz poluçennoho protyvore-
çyq sleduet, çto pry uslovyqx (2.1) y (2.2) dolΩno b¥t\ λ0 ∈Λk .
Teorema dokazana.
Otmetym, çto pry reßenyy matryçnoho neravenstva (2.1) net nykakyx ohra-
nyçenyj na matrycu B. V çastnosty, ona moΩet b¥t\ nulevoj. V sluçae k > s
teoremaC2.1 soxranqet sylu, esly ne vse bloky As+1 , … , Am matryc¥ A nule-
v¥e, no takov¥, çto spektr matryçnoho polynoma Fm ( )λ = A0 + λ A1 + …
… + λm
mA soderΩyt yzuçaem¥j spektr σ( )F . V sluçae k ≤ s dann¥e bloky v
A otsutstvugt. Matrycu H celesoobrazno yskat\ v vyde poloΩytel\no
opredelennoj matryc¥, naprymer moΩno poloΩyt\ H In= . Esly uslovyq
teorem¥C2.1 v¥polnqgtsq pry H ≤ 0, to vse sobstvenn¥e znaçenyq
vspomohatel\noho matryçnoho polynoma G( )λ tak Ωe, kak y F( )λ , dolΩn¥
prynadleΩat\ oblasty (1.3). Ohranyçenye (2.2) na bloçnug matrycu X
v¥polnqetsq dlq lgboj oblasty Λk v sluçae, kohda Ω — vsq kompleksnaq
ploskost\, t.Ce. Z XZr r( ) ( )λ λ∗ ≥ 0 ∀ λ ∈ C. Dlq πtoho dostatoçno potrebovat\,
çtob¥ yskomaq matryca X b¥la neotrycatel\no opredelennoj.
TeoremaC2.1 obobwaet y razvyvaet metod lokalyzacyy spektra matryçnoho
polynoma, predloΩenn¥j v [6] dlq klassa oblastej Λ1 v vyde matryçn¥x ne-
ravenstv (1.2). Esly k = 1 y H = 0, to matryçn¥e neravenstva (2.1) y (1.2) sov-
padagt. V dannom utverΩdenyy net ohranyçenyj na porqdok yspol\zuem¥x al-
hebrayçeskyx kryv¥x, a uslovye (2.2) v obwem sluçae ne trebuet poloΩytel\noj
opredelennosty reßenyj X sootvetstvugwyx matryçn¥x neravenstv.
Sleduet obratyt\ vnymanye na sluçaj k ≥ s, pryvodqwyj k sledstvyg teo-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
LOKALYZACYQ SOBSTVENNÁX ZNAÇENYJ POLYNOMYAL|NÁX MATRYC 1067
rem¥C2.1 v vyde teorem¥C1.1. V πtom sluçae m = k, r = 0, C = +In k( )1 y matryç-
noe neravenstvo (2.1) uprowaetsq k vydu (1.4). Pry πtom neyzvestn¥e πrmytov¥
matryc¥ X y H ymegt odynakov¥e razmer¥ n × n y v teoremeC1.1 vmesto uslo-
vyq (2.2) yspol\zuem neravenstvo X = X∗ ≥ 0.
Pust\ A+ = ( )A A A∗ − ∗1
— psevdoobratnaq matryca, a A⊥
— matryca raz-
mera n m( )+ 1 × n m, sostavlennaq yz ortohonal\noho dopolnenyq lynejno ne-
zavysym¥x vektor-stolbcov A do bazysa prostranstva Cn m( )+1
. Tohda
A A+ = In , A A⊥∗ = 0nm n, , det T ≠ 0 , T =
+∗ ⊥A A, . (2.3)
UmnoΩym sleva (sprava) matryçnoe neravenstvo (2.1) na T ∗
(T) y yspol\zuem
yzvestn¥j kryteryj poloΩytel\noj opredelennosty bloçnoj matryc¥:
P V
V Q∗
> 0 ¤ Q > 0, P > VQ V− ∗1 . (2.4)
V dannom sluçae
P = H + B A∗ +∗ + A B+ + A A+ +∗L X( ) , Q = A A⊥∗ ⊥L X( ) ,
V = B A A∗ + ⊥+ L X( ) .
Tak kak poslednee neravenstvo v (2.4) vsehda moΩno udovletvoryt\ putem
v¥bora matryc¥ H > 0, razreßymost\ neravenstva Q > 0 πkvyvalentna razre-
ßymosty ysxodnoho neravenstva (2.1). Poπtomu spravedlyvo sledugwee ut-
verΩdenye.
Teorema 2.2. Pust\ dlq nekotoroj matryc¥ X X= ∗
razmera n m( )+ 1 ×
× n m( )+ 1 v¥polnqgtsq uslovye (2.2) y matryçnoe neravenstvo
M X D X Dij i j
i j
k
( ) :
,
= >∗
=
∑ γ
0
0 , (2.5)
hde Di = A⊥∗Ci , i = 0, … , k . Tohda vse sobstvenn¥e znaçenyq matryçnoho po-
lynoma (1.1) raspoloΩen¥ v oblasty (1.3).
Otmetym, çto esly det A0 0≠ , to matrycu A⊥∗
, udovletvorqgwug soot-
noßenyqm (2.3), vsehda moΩno postroyt\ v vyde
A⊥∗ =
ˆ
ˆ
,
,
A I
A I
n n n
m n n n
1 0
0
�
� � � �
�
, ˆ , ,
, .,
A
A A i s
i s
i
i
n n
=
− ≤
>
−
0
1
esly
0 esly
(2.6)
V sluçae k = s operator (2.5) s uçetom (2.6) ymeet bloçnug strukturu
M X( ) = Y Zpq p q
s
( )
, =1
> 0, (2.7)
hde Y Zpq ( ) = γ 00 A ZAp q
∗ – γ p qA Z A0 0
∗ – γ 0 0q pA Z A∗ + γ pq A Z A0 0
∗ , X A Z A= ∗
0 0 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
1068 A. H. MAZKO
Pry uslovyqx rehulqrnosty matryçnoho polynoma F( )λ suwestvuet α ∈C
takoe, çto det Aα0 0≠ , hde
A Fα α0 = ( ) , F Fα λ λ α( ) : ( )= + = Aα0 + λ αA 1 + … + λ α
s
sA .
Poskol\ku σ α( )F = σ( )F – α, vklgçenyq σ( )F Ã Λk y σ α( )F Ã Λαk πkvy-
valentn¥. Zdes\ oblast\ Λαk opredelena v vyde (1.3) s yspol\zovanyem πrmy-
tovoj matryc¥ Γα , sostavlennoj yz koπffycyentov razloΩenyq
f ( , )λ α λ α+ + = γ λ λα
ij
i j
i j
k
( )
, =
∑
0
.
Otmetym, çto suwestvugt razlyçn¥e sposob¥ svedenyq spektral\n¥x zadaç
dlq matryçn¥x polynomov k analohyçn¥m zadaçam dlq lynejn¥x puçkov mat-
ryc. Pryvedem odyn yz nyx, osnovann¥j na prymenenyy matryc typa A⊥ [7].
Yspol\zovav bloçnoe predstavlenye matryc¥ A⊥∗ = U Us0, ,…[ ] , udovlet-
vorqgwej sootnoßenyqm (2.3), postroym lynejn¥j puçok matryc
A B− λ = U Us1, ,…[ ] – λ U Us0 1, ,…[ ]− (2.8)
y rassmotrym sledugwye sootnoßenyq:
u F∗ =( )λ 0 , u ≠ 0 , (2.9)
v∗ − =( )A Bλ 0 , v ≠ 0 , (2.10)
v∗ ⊥∗A = u Zs
∗ ( )λ , u = U0
∗v ≠ 0. (2.11)
Sootnoßenyq (2.9) y (2.10) opredelqgt sobstvenn¥e znaçenyq y sootvetst-
vugwye lev¥e sobstvenn¥e vektor¥ matryçnoho polynoma (1.1) y puçka matryc
(2.8). Lehko ustanovyt\ πkvyvalentnost\ sootnoßenyj (2.10) y (2.11). S druhoj
storon¥, yz opredelenyq matryc¥ A⊥∗
y predstavlenyq F( )λ = Zs ( )λ A
sleduet, çto v¥polnenye sootnoßenyj (2.9) ravnosyl\no v¥polnenyg sootno-
ßenyj (2.11) dlq nekotoroho vektora v ≠ 0 . Sledovatel\no, sootnoßenyq (2.9)
y (2.10) πkvyvalentn¥. Pry πtom toΩdestvennoe v¥polnenye odnoho yz ravenstv
(2.9) yly (2.10) pry lgbom λ ∈C vleçet toΩdestvennoe v¥polnenye druhoho
yz nyx. ∏to oznaçaet, çto svojstva rehulqrnosty matryçnoho polynoma (1.1) y
puçka matryc (2.8) takΩe πkvyvalentn¥.
Lemma 2.1. MnoΩestva vsex razlyçn¥x sobstvenn¥x znaçenyj λi matryç-
noho polynoma (1.1) y puçka matryc (2.8) sovpadagt, a yx sootvetstvugwye
lev¥e sobstvenn¥e vektor¥ svqzan¥ sootnoßenyqmy u Ui i
∗ ∗= v 0 , i = 1, … , l.
Sformulyruem kryteryy prynadleΩnosty spektra matryçnoho polynoma
F( )λ oblastqm klassa Λ1 . V πtom sluçae operator (2.5) svodytsq k vydu
M X( ) = γ 00BX B∗ + γ 01BX A∗ + γ10 AX B∗ + γ11AX A∗ , (2.12)
hde A y B — matryc¥ lynejnoho puçka (2.8). Dejstvytel\no, yspol\zuq bloç-
nug strukturu matryc
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LOKALYZACYQ SOBSTVENNÁX ZNAÇENYJ POLYNOMYAL|NÁX MATRYC 1069
A⊥∗ = …[ ]U Us0, , , E
Is
s
=
01,
, ∆ =
0 0
0
1
1
,
,
s
s sI
,
poluçaem matryçn¥e koπffycyent¥
D0 = A⊥∗ ⊗( )E In = B, D1 = A⊥∗ ⊗( )∆E In = A.
V çastnosty, dlq matryc¥ (2.6) ymeem
B
A I
A I
A
n n n
s n n n
s n n n n
=
−
ˆ
ˆ
ˆ
,
,
, ,
1
1
0
0
0 0
�
� � � �
�
�
, A Ins= .
Uçyt¥vaq lemmuC2.1 y obwye svojstva poloΩytel\no obratym¥x operatorov
v prostranstve πrmytov¥x matryc, moΩno sformulyrovat\ sledugwee utverΩ-
denye (sm., naprymer, [3]).
Teorema 2.3. Vklgçenye σ( )F ⊂ Λ1 v¥polnqetsq v tom y tol\ko v tom
sluçae, kohda suwestvugt πrmytov¥ matryc¥ X y Y , udovletvorqgwye
sootnoßenyqm:
1) M X( ) = Y ≥ 0, BX B∗ ≥ 0, rank A B Y−[ ]λ , ≡ ns.
Esly matryca B nev¥roΩdena, to dannoe vklgçenye πkvyvalentno kaΩdo-
mu yz sledugwyx utverΩdenyj:
2) matryçnoe neravenstvo M X( ) > 0 ymeet reßenye X X= ∗ > 0;
3) dlq lgboj matryc¥ Y Y= ∗ > 0 matryçnoe uravnenye M X( ) = Y yme-
et reßenye X X= ∗ > 0;
4) operator M poloΩytel\no obratym otnosytel\no konusa neotryca-
tel\no opredelenn¥x matryc.
Dokazatel\stvo dostatoçnosty kryteryq 1 sostoyt v sledugwem. Esly
predpoloΩyt\, çto nekotoroe sobstvennoe znaçenye λ0 1∉Λ , to sohlasno
(2.10) dlq sootvetstvugweho levoho sobstvennoho vektora v∗
dolΩn¥ v¥pol-
nqt\sq protyvoreçyv¥e sootnoßenyq
f ( , )λ λ0 0 ≤ 0, v v∗ ∗BX B ≥ 0, f BX B( , )λ λ0 0 v v∗ ∗ = v v∗Y > 0.
Pry uslovyy σ( )F ⊂ Λ1 matryc¥ X y Y v kryteryy 1 vsehda moΩno postroyt\
v vyde [3]
X Z X Z= ∗ˆ , Y BZY Z B= ∗ ∗ˆ , rank rankAZ BZ BZ, ( )[ ] = .
Poslednyj fakt ustanavlyvaetsq s yspol\zovanyem kanonyçeskoj form¥ Kro-
nekera rehulqrnoho puçka matryc [8]. V sluçae nev¥roΩdennoj matryc¥ B
kryteryy 2 – 4 sledugt yz yzvestn¥x teorem o lokalyzacyy sobstvenn¥x znaçe-
nyj s pomow\g obobwennoho uravnenyq Lqpunova.
Zameçanye 2.1. Dostatoçn¥e uslovyq vklgçenyq σ( )F k⊂ Λ , predstav-
lenn¥e teoremamy 2.1 y 2.2, pry opredelenn¥x ohranyçenyqx mohut stat\ y ne-
obxodym¥my ne tol\ko v sluçae k = 1 (teoremaC2.3). MoΩno pokazat\, çto pry
v¥polnenyy matryçnoho neravenstva (2.1) i L X+( )( ) ≥ n m y, sledovatel\no (sm.
[4], teorema 4.2.1),
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
1070 A. H. MAZKO
i i X+ +( ) ( )Γ + i i X− −( ) ( )Γ ≥ n m.
Poslednee neravenstvo neobxodymo takΩe dlq v¥polnenyq uslovyj teore-
m¥C2.2. V çastnosty, esly k ≥ s y X > 0, to dlq v¥polnenyq matryçnoho nera-
venstva (2.5) neobxodymo ohranyçenye i( )Γ = k, ,1 0{ } . Kryteryy razreßymosty
matryçnoho neravenstva (2.5) mohut b¥t\ yzuçen¥ na osnove obwyx teorem ob
ynercyy πrmytov¥x reßenyj transformyruem¥x matryçn¥x uravnenyj [3, 4], v
kotor¥x yspol\zugtsq ohranyçenyq na ynercyg matryc¥ Γ y svojstva odno-
vremennoj pryvodymosty k treuhol\nomu vydu matryçn¥x koπffycyentov Di ,
i = 0, … , k.
3. Prymer¥. 1. Pust\ F( )λ = A – λ B — rehulqrn¥j puçok n × n-matryc y
Λ1 — oblast\ vyda (1.3). Matryçnoe neravenstvo (2.1) ymeet vyd
AB B A AHA X AB B B AHB X
BB
0 0 00 1 0 01
0
∗ ∗ ∗ ∗ ∗ ∗
∗
+ + + − − +
−
γ γ
++ − + − − + +
∗ ∗ ∗ ∗ ∗B A BHA X BB B B BHB X1 10 1 1 11γ γ
> 0.
Matryçnoe neravenstvo (2.5) s yspol\zovanyem operatora M (X) dlq lynejnoho
puçka Fα λ( ) = F( )α – λ B y oblasty Λα1 = λ ∈{ C : f ( , )λ α λ α+ + > 0} ,
pryvodytsq k vydu
γ 00BZB∗ + γ10 AZB∗ + γ 01BZA∗ + γ11AZA∗ > 0,
hde X = F ZF( ) ( )α α∗
, α σ∉ ( )F . Pry πtom A⊥∗ = BF In
−
1( ),α . Dannoe nera-
venstvo v sylu πkvyvalentnosty vklgçenyj σ( )F ⊂ Λ1 y σ α α( )F ⊂ Λ 1 moΩno
yspol\zovat\ v teoremeC2.2 dlq ysxodnoho puçka matryc F( )λ y oblasty Λ1 . V
çastnosty, polahaq
Γ =
−
−
0 1
1 0
y Γ =
−
1 0
0 1
, (3.1)
ymeem uslovyq razmewenyq spektra σ( )F vnutry levoj poluploskosty y edy-
nyçnoho kruha v vyde suwestvovanyq reßenyj Z = Z∗ > 0 sootvetstvugwyx
matryçn¥x neravenstv
AZ B BZA∗ ∗+ < 0 y BZB AZA∗ ∗− > 0.
2. Pust\ F( )λ = A + λ B+ λ2C — rehulqrn¥j kvadratyçn¥j puçok n × n-
matryc y Λk , k ≤ 2 — oblast\ vyda (1.3). V¥raΩenye AB∗ + BA∗ + A AH ∗
v
(2.1) ymeet vyd
AB B A AHA AB B B AHB AB B C AHC
B
0 0 1 0 2 0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗+ + + + + +
BB B A BHA BB B B BHB BB B C BHC
C
0 1 1 1 2 1
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗+ + + + + +
BB B A CHA CB B B CHB CB B C CHC0 2 1 2 2 2
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗+ + + + + +
.
Dlq klassa oblastej Λ1 ymeem 1 = k < s = 2, m = 2, r = 1,
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LOKALYZACYQ SOBSTVENNÁX ZNAÇENYJ POLYNOMYAL|NÁX MATRYC 1071
E =
1 0
0 1
0 0
, ∆ =
0 0 0
1 0 0
0 1 0
, R =
1 0 0 0
0 1 1 0
0 0 0 1
,
X
X X
X X
=
00 01
10 11
,
L X( ) =
γ γ γ γ
γ γ γ
00 00 00 01 01 00 01 01
00 10 10 00 1
X X X X
X X Xij
+
+ −− −
=
∑ +
+
i j
ij
X X
X X X
1
0
1
01 11 11 01
10 10 10 11 11 1
γ γ
γ γ γ 00 11 11γ X
.
Dlq klassa oblastej Λ2 , ohranyçenn¥x alhebrayçeskymy kryv¥my ne v¥ße 4-
ho porqdka, v teoremax 2.1 y 2.2 yspol\zuem operator¥
L X( ) =
γ γ γ
γ γ γ
γ γ γ
00 01 02
10 11 12
20 21 22
X X X
X X X
X X X
, M X( ) =
Y Z Y Z
Y Z Y Z
11 12
21 22
( ) ( )
( ) ( )
,
hde
Y Z11( ) = γ 00BZB∗ – γ10 AZB∗ – γ 01BZA∗ + γ11AZA∗ ,
Y Z12( ) = γ 00BZC∗ – γ10 AZC∗ – γ 02BZA∗ + γ12 AZA∗ ,
Y Z22( ) = γ 00CZC∗ – γ 20 AZC∗ – γ 02CZA∗ + γ 22 AZA∗ .
3. Pust\ F( )λ = a0 + λa1 + … + λs
sa — skalqrn¥j polynom stepeny s.
Matryçnoe neravenstvo (2.1) ymeet vyd
ab ba haa L X∗ ∗ ∗+ + + ( ) > 0,
hde a s∈ +C 1
— vektor koπffycyentov polynoma F( )λ , h ∈R , b s∈ +C 1
. Pry
opredelenyy operatora L X( ) dlq klassa oblastej Λ1 yspol\zuem matryc¥
E
Is
s
=
01,
, ∆ =
0 0
0
1
1
,
,
s
s sI
, C =
I
I
s
s
s
s0
0
1
1
,
,
,
X xij i j
s=
=
−
, 0
1
.
V çastnosty, dlq levoj poluploskosty y edynyçnoho kruha, otveçagwyx matry-
cam (3.1), dann¥j operator ymeet sootvetstvenno vyd
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1072 A. H. MAZKO
L X( ) = −
+ +
− −
− − −
0 00 0 2 0 1
00 01 10 0 1 1 2 1 1
x x x
x x x x x x
s s
s s s
�
�
�� � � � �
�x x x x x xs s s s s s s s s− − − − − − − −+ +20 10 21 1 2 2 1 1 −−
− − − −
1
10 11 1 1 0x x xs s s s�
,
L X( ) =
x x x
x x x x x x
s
s s s
00 01 0 1
10 11 00 1 1 0 2 0 1
0�
�
� �
−
− − −− − −
�� � �
�x x x x x xs s s s s s s s s− − − − − − − − −− − −10 11 20 1 1 2 2 2 11
10 1 2 1 10 − − −
− − − − −x x xs s s s s�
.
Dlq kaΩdoj oblasty Λk pry k ≥ s operator L X( ) = x Γ qvlqetsq matryç-
noznaçnoj funkcyej skalqrnoho arhumenta X = x.
4. Semejstva matryçn¥x polynomov. Rassmotrym parametryçeskoe se-
mejstvo rehulqrn¥x matryçn¥x polynomov
F p( , )λ = A p0 0( ) + λ A p1 1( ) + … + λs
s sA p( ) , det ( , )F pλ /≡ 0 . (4.1)
Znaçenyq matryçn¥x koπffycyentov A pi i( ) , zavysqwyx ot parametrov
p p pi i i
T
i
= … 1, , ν ∈ Pν
ν
ν
ν
j
i
i
i
q q q q j
j
: : , , ,= ∈ ≥ … ≥ =
=
∑R 1
1
0 0 1
,
obrazugt semejstvo polytopov n × n-matryc
Ai = A C A p A pn n
ij ij
j
i
i
i
∈ = ∈
×
=
∑: ,
1
ν
νP , i = 0, … , s.
Obwyj vektor parametrov p = p pT
s
T T
0 , ,… ∈ P = Pν0
× … × Pνs
ymeet porq-
dok ν = ν0 + … + νs .
V¥delym vse „verßynn¥e” matryçn¥e polynom¥
Ft ts0 … ( )λ = A t0 0
+ λ A t1 1
+ … + λs
stA
s
, ti i∈ …{ }1, , ν , i = 0, … , s.
Yx kolyçestvo ravno ν0 … νs . Esly νi = 1 dlq nekotoroho i, to ti = 1 y v
(4.1) sootvetstvugwyj matryçn¥j koπffycyent Ai ne zavysyt ot pi .
Lemma 4.1. Dlq lgboho vektora p ∈Pν v¥polnqetsq matryçnoe nera-
venstvo ppT ≤ P : = diag ( , , )p p1 … ν .
Dokazatel\stvo. PokaΩem, çto dlq lgboho vektora x ∈Rν
v¥polnqetsq
neravenstvo ψ( )x = x P pp xT T( )− ≥ 0. Dejstvytel\no,
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LOKALYZACYQ SOBSTVENNÁX ZNAÇENYJ POLYNOMYAL|NÁX MATRYC 1073
ψ( )x = p xi i
i
2
1=
∑
ν
– p xi i
i=
∑
1
2ν
= p x pi i
i
j
j
2
1 1= =
∑ ∑
ν ν
– p xi i
i=
∑
1
2ν
=
= p p xi j i
i j
2
≠
∑ – p p x xi j i j
i j≠
∑ = p p x xi j i j
i j
( )−
<
∑ 2 ≥ 0.
Lemma dokazana.
Teorema 4.1. Pust\ dlq nekotor¥x matryc B, H = H∗ ≤ 0 y Xt ts0 … =
= Xt ts0 …
∗
v¥polnqgtsq uslovye (2.2) y systema matryçn¥x neravenstv
A Bt ts0 …
∗ + BAt ts0 …
∗ + A At t t ts s
H
0 0… …
∗ + L Xt ts
( )
0 … > 0, (4.2)
hde
At ts0 … =
A
Am
0
�
, A
A i s
i s
i
it
n n
i=
≤
>
, ,
, ,,
esly
esly0
ti i∈ …{ }1, , ν , i = 0, … , s.
Tohda pry lgbom p ∈P vse sobstvenn¥e znaçenyq matryçnoho polynoma (4.1)
raspoloΩen¥ v oblasty (1.3).
Dokazatel\stvo. PokaΩem, çto dlq lgboho p ∈P v¥polnqetsq matryç-
noe neravenstvo
A B( )p ∗ + BA( )p ∗ + A A( ) ( )p H p ∗ + L X p( )( ) > 0, (4.3)
hde
A( )p =
A
Am
0
�
, A
A p i s
i s
i
i i
n n
=
≤
>
( ), ,
, ,,
esly
esly0
ti i∈ …{ }1, , ν , i = 0, … , s,
X p( ) — neotrycatel\naq lynejnaq kombynacyq matryc Xt ts0 … , udovletvo-
rqgwaq uslovyg (2.2).
Vse matryçn¥e neravenstva system¥ (4.2) uporqdoçyvagtsq naboramy yndek-
sov t ts0, ,…{ } . Zafyksyruem yndeks¥ i = 0 y t j ∈ 1, ,…{ }ν j , j ≠ 0. Zatem
umnoΩym ν0 neravenstv (4.2), otveçagwyx naboram yndeksov 1{ , t1 , …
…, ts} ,C… , ν0 1, , ,t ts…{ } , sootvetstvenno na p01, … , p0 0ν y prosummyruem
yx, uçyt¥vaq, çto p0 = p p
T
01 0 0
, ,… ν ∈ Pν0
. Dlq vsevozmoΩn¥x kombyna-
cyj yndeksov t j ∈ 1, ,…{ }ν j , j ≠ 0, v¥polnym takye Ωe operacyy. Poluçen-
n¥e neravenstva budem yspol\zovat\ v systeme (4.2) vmesto uΩe rassmotrenn¥x
neravenstv. Pry πtom vse neravenstva dannoj system¥ uporqdoçyvagtsq nabo-
ramy yndeksov t ts1, ,…{ } , hde t j ∈ 1, ,…{ }ν j , j = 1, … , s. Analohyçnug pro-
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1074 A. H. MAZKO
ceduru v¥polnym pry kaΩdom i = 1, … , s , yspol\zovav vektor¥ p1 =
= pi i
T
p
i1, ,… ν ∈ Pνi
. Na poslednem ßahe poluçym odno bloçnoe neravenstvo,
k perv¥m s + 1 dyahonal\n¥m blokam kotoroho prybavym y v¥çtem sootvetst-
vugwye v¥raΩenyq A p HA pi i i i( ) ( )∗
. V rezul\tate poluçym matryçnoe nera-
venstvo
A B( )p ∗ + BA( )p ∗ + A A( ) ( )p H p ∗ + S p( ) + L X p( )( ) > 0,
hde S p( ) — bloçno-dyahonal\naq matryca s dyahonal\n¥my blokamy
S
p A HA A p HA p i s
i
ij ij ij i i i i
j
i
=
− ≤∗ ∗
=
∑ ( ) ( ) ,
1
ν
esly ,,
, ,,0n n i sesly >
i = 1, … , m.
Predstavym perv¥e s dyahonal\n¥x blokov dannoj matryc¥ v vyde
Si = W P p p H Wi i i i
T
i−( ) ⊗
∗ , Wi = A Ai i i1, ,… ν ,
Pi = diag ( , , )p pi i i1 … ν .
Sohlasno lemmeC4.1 Pi ≥ p pi i
T
pry pi i∈P . Yspol\zuq sledugwee svojstvo
kronekerovoho proyzvedenyq πrmytov¥x matryc:
P = P∗ ≥ 0, Q = Q∗ ≤ 0 fi P Q⊗ ≤ 0,
poluçaem, çto S p( ) ≤ 0 pry p ∈P . Sledovatel\no, v¥polnqetsq matryçnoe
neravenstvo (4.3) y, sohlasno teoremeC2.1, vse sobstvenn¥e znaçenyq matryçnoho
polynoma (4.1) raspoloΩen¥ v oblasty (1.3) dlq lgboho p ∈P .
Teorema dokazana.
Zameçanye 4.1. Esly systema matryçn¥x neravenstv (4.2) v¥polnqetsq pry
H = H∗ ≤ 0, to ona v¥polnqetsq y pry H = 0. Poπtomu pry yspol\zovanyy teo-
rem¥C4.1 vsehda moΩno poloΩyt\ H = 0. Esly H = H∗ < 0, to matryçnoe nera-
venstvo (4.3) πkvyvalentno bloçnomu neravenstvu
A B BA A
A
( ) ( ) ( ) ( )
( )
p p L X p p
p H
∗ ∗
∗ −
+ + ( )
−
1
> 0. (4.4)
V analohyçnom vyde predstavlqgtsq vse neravenstva system¥ (4.2). Zavysy-
most\ ot parametrov p v (4.4) lynejna. Poπtomu v dannom sluçae moΩno pro-
vesty dokazatel\stvo teorem¥C4.1 bez yspol\zovanyq lemm¥C4.1.
Otmetym, çto teoremuC4.1 moΩno yspol\zovat\ dlq parametryçeskyx y yn-
terval\n¥x semejstv rehulqrn¥x matryçn¥x polynomov vyda
F p( , )λ = p A A Ai i i
s
si
i
( )0 1
1
+ + … +
=
∑ λ λ
ν
, p ∈Pν , (4.5)
F( )λ = A0 + λ A1 + … + λs
sA , A A Ai i i≤ ≤ , i = 0, … , s. (4.6)
Semejstvo (4.1) pryvodytsq k vydu (4.5) v sluçae, kohda vse vektor¥ parametrov
pi i
∈Pν ymegt odynakov¥j porqdok ν y sovpadagt. V πtom sluçae syste-
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LOKALYZACYQ SOBSTVENNÁX ZNAÇENYJ POLYNOMYAL|NÁX MATRYC 1075
maC(4.2) sostoyt yz ν matryçn¥x neravenstv. Ynterval\noe semejstvo (4.6),
naybolee çasto yspol\zuemoe v pryloΩenyqx, takΩe opys¥vaetsq v vyde (4.1).
Dejstvytel\no, dlq πtoho v (4.1) sleduet poloΩyt\
Ai = A pi i( ) = p Aij ij
j
i
=
∑
1
ν
, A aij t
ij
t
n
=
=τ τ, 1
, a a at
ij
t
i
t
i
τ τ τ∈{ }, , νi
n= 2
2
,
A ai t
i
t
n
=
=τ τ, 1
, A ai t
i
t
n
=
=τ τ, 1
, pi = p p Ei i
T
i1, ,… ν ∈ Pνi
,
i = 0, … , s.
Pry πtom systema (4.2) budet sostoqt\ yz 2 1 2( )s n+
matryçn¥x neravenstv.
Prymer 4.1. Rassmotrym dvuxmassovug mexanyçeskug systemu, yzobraΩen-
nug na rys. 1 y opys¥vaemug uravnenyqmy [9]
m x1 1�� + d x1 1� + ( )c c x1 12 1+ – c x12 2 = 0,
(4.7)
m x2 2�� + d x2 2� + ( )c c x2 12 2+ – c x12 1 = u.
Rys. 1. Mexanyçeskaq systema.
Rys. 2. Oblast\ ustojçyvosty Λ2 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
1076 A. H. MAZKO
V [6] proveden analyz robastnoj ustojçyvosty dannoj system¥ s ynterval\-
noj neopredelennost\g parametrov a a≤ = c1[ , c2 , d1 , d2 , m1 , m2] ≤ a y
trebovanyem razmewenyq spektra vnutry kruha radyusa 12 s centrom v toçke
(– 12, 0).
Opredelym v levoj poluploskosty oblast\
Λ2 = λ λ λ∈ >{ }∗C : ( ) ( )z zΓ 0 , (4.8)
hde
z( )λ = 1 2, ,λ λ , Γ = −
9
16
7
4
9
4
7
4
3 3
9
4
3 1
4 3 2
3 2
2
α α α
α α α
α α
, α > 0.
Hranycej dannoj oblasty qvlqetsq kryvaq 4-ho porqdka, naz¥vaemaq ulytkoj
Paskalq (sm. rys. 2), uravnenye kotoroj ymeet vyd
( ) ( )x h y x h+ + + +
2 2 2
2 α – β2 2 2( )x h y+ + = 0,
hde β = 2α, h = α / 2. PoloΩym
α = 1, 3, a = 5 6 6 9 2 4, , , , ,[ ] , a = 6 7 7 10 4 7, , , , ,[ ] , c12 1= .
Tohda ynterval\n¥j matryçn¥j polynom (4.6), sootvetstvugwyj razomknutoj
systeme, ymeet vyd F( )λ = A0 + λ A1 + λ2
2A , hde
A0 =
6 1
1 7
−
−
≤ A0 =
c
c
1
2
1 1
1 1
+ −
− +
≤ A0 =
7 1
1 8
−
−
,
A1 =
6 0
0 9
≤ A1 =
d
d
1
2
0
0
≤ A1 =
7 0
0 10
,
A2 =
2 0
0 4
≤ A2 =
m
m
1
2
0
0
≤ A2 =
4 0
0 7
.
Systema (4.2) s neyzvestn¥my Xt t t0 1 2
, t0 , t1 , t2 ∈ 1 4, ,…{ } , B y H sostoyt yz
64 matryçn¥x neravenstv. Pry Bi = ( )A Ai i+ / 2, i = 0, 1, 2, y H = 0 s pomo-
w\g system¥ MATLAB ustanovleno, çto πta systema ymeet reßenye X = X∗ >
> 0. Sledovatel\no, systema (4.7) robastno ustojçyva y vse ee sobstvenn¥e zna-
çenyq naxodqtsq v oblasty (4.8) pry ukazann¥x ynterval\n¥x neopredelennos-
tqx. Na rys. 2 pokazano raspoloΩenye vsex sobstvenn¥x znaçenyj 64 kvadra-
tyçn¥x puçkov matryc Ft t t0 1 2
( )λ , t0 , t1 , t2 ∈ 1 4, ,…{ } , v oblasty Λ2 .
V zaklgçenye otmetym, çto yzloΩennaq metodyka lokalyzacyy sobstvenn¥x
znaçenyj matryçn¥x polynomov moΩet b¥t\ rasprostranena na nekotor¥e
klass¥ matryçn¥x funkcyj vyda
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
LOKALYZACYQ SOBSTVENNÁX ZNAÇENYJ POLYNOMYAL|NÁX MATRYC 1077
F( )λ = f Ai i
i
s
( )λ
=
∑
0
, det ( )F λ /≡ 0 ,
hde Ai — postoqnn¥e n × n -matryc¥, fi ( )λ — skalqrn¥e funkcyy,
analytyçeskye v okrestnosty toçek spektra σ( )F . Pry πtom oblasty lokaly-
zacyy sobstvenn¥x znaçenyj v kompleksnoj ploskosty moΩno opys¥vat\ v vyde
Λ = ∈ ( ) ≥{ })+λ λ λC : ( , )i V 1 , V ( , )λ λ = f fi j ij
i j
s
( ) ( )
,
λ λ Γ
=
∑
0
,
hde Γ ij — bloky πrmytovoj matryc¥ Γ. Takoe obobwenye, v çastnosty, daet
vozmoΩnost\ poluçyt\ alhebrayçeskye uslovyq absolgtnoj ustojçyvosty ly-
nejn¥x dyfferencyal\n¥x system s zapazd¥vanyem, opys¥vaem¥x s pomow\g
matryçn¥x kvazypolynomov.
1. Gohberg I., Lancaster P., Rodman L. Matrix polynomials. – New York: Acad. Press, 1982. – xiv +
409 p.
2. Markus A. S. Vvedenye v spektral\nug teoryg polynomyal\n¥x operatorn¥x puçkov. –
Kyßynev: Ítyynca, 1986. – 259 s.
3. Mazko A. H. Lokalyzacyq spektra y ustojçyvost\ dynamyçeskyx system // Pr. In-tu matema-
tyky NAN Ukra]ny. – 1999. – 28. – 216 s.
4. Mazko A. G. Matrix equations, spectral problems and stability of dynamic systems // Stability,
Oscillations and Optimization of Systems / Eds A. A. Martynyuk, P. Borne, and C. Cruz-
Hernandez. – Cambridge: Cambridge Sci. Publ., 2008. – Vol. 2. – xx + 270 p.
5. Henrion D., Bachelier O., Sebek M. D-stability of polynomial matrices // Int. J. Control. – 2001. –
74, # 8. – P. 355 – 361.
6. Henrion D., Arzelier D., Peaucelle D. Positive polynomial matrices and improved LMI robustness
conditions // Automatica. – 2003. – 39, # 8. – P. 1479 – 1485.
7. Kublanovskaq V. N. K spektral\noj zadaçe dlq polynomyal\n¥x puçkov matryc // Zap.
nauç. sem. LOMY. – 1978. – 80. – S. 83 – 97.
8. Hantmaxer F. R. Teoryq matryc. – M.: Nauka, 1988. – 552 s.
9. Ackermann J. Robust control. Systems with uncertain physical parameters. – London: Springer,
1993. – xvi + 406 p.
Poluçeno 29.10.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
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| id | umjimathkievua-article-2937 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:33:09Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dc/5f8913bc4965162511a37f9e8eb420dc.pdf |
| spelling | umjimathkievua-article-29372020-03-18T19:40:46Z Localization of eigenvalues of polynomial matrices Локализация собственных значений полиномиальных матриц Mazko, A. G. Мазко, А. Г. Мазко, А. Г. We consider the problem of localization of eigenvalues of polynomial matrices. We propose sufficient conditions for the spectrum of a regular matrix polynomial to belong to a broad class of domains bounded by algebraic curves. These conditions generalize the known method for the localization of the spectrum of polynomial matrices based on the solution of linear matrix inequalities. We also develop a method for the localization of eigenvalues of a parametric family of matrix polynomials in the form of a system of linear matrix inequalities. Роботу присвячено задачі локалізації власних значень поліноміальиих матриць. Запропоновано достатні умови належності спектра регулярного матричного полінома широкому класу областей, обмежених алгебраїчними кривими. Ці умови узагальнюють відомий метод локалізації спектра поліноміальиих матриць, що зводи ться до розв'язання лінійних матричних нерівностей. Розвивається метод локалізації власних значень параметричної сім'ї матричних поліномів у вигляді системи лінійних матричних нерівностей. Institute of Mathematics, NAS of Ukraine 2010-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2937 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 8 (2010); 1063–1077 Український математичний журнал; Том 62 № 8 (2010); 1063–1077 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2937/2624 https://umj.imath.kiev.ua/index.php/umj/article/view/2937/2625 Copyright (c) 2010 Mazko A. G. |
| spellingShingle | Mazko, A. G. Мазко, А. Г. Мазко, А. Г. Localization of eigenvalues of polynomial matrices |
| title | Localization of eigenvalues of polynomial matrices |
| title_alt | Локализация собственных значений полиномиальных матриц |
| title_full | Localization of eigenvalues of polynomial matrices |
| title_fullStr | Localization of eigenvalues of polynomial matrices |
| title_full_unstemmed | Localization of eigenvalues of polynomial matrices |
| title_short | Localization of eigenvalues of polynomial matrices |
| title_sort | localization of eigenvalues of polynomial matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2937 |
| work_keys_str_mv | AT mazkoag localizationofeigenvaluesofpolynomialmatrices AT mazkoag localizationofeigenvaluesofpolynomialmatrices AT mazkoag localizationofeigenvaluesofpolynomialmatrices AT mazkoag lokalizaciâsobstvennyhznačenijpolinomialʹnyhmatric AT mazkoag lokalizaciâsobstvennyhznačenijpolinomialʹnyhmatric AT mazkoag lokalizaciâsobstvennyhznačenijpolinomialʹnyhmatric |