Iden ical pseudospec a o any geome ic
mul iplici y∗
Go ka A men ia†
, Juan-Miguel G acia‡
, F ancisco E. Velasco‡
Janua y 9, 2011
Dedica ed o P o esso Jos´e An ´onio Dias da Sil a
Abs ac
I A, B a e n×ncomplex ma ices such ha he singula alues o
zIn−Aa e he same as hose o zIn−B o each z∈C, hen Aand B
a e simila .
AMS classi ica ion: 15A18, 15A21, 15A60, 47A25.
Key Wo ds: singula alues, simila i y, pseudospec um, in ini esimals.
1 In oduc ion
Le M∈Cn×n. Le Λ(M) deno e he spec um o Mand le σ1(M)≥
σ2(M)≥ · · · ≥ σn(M) deno e he singula alues o Ma anged in dec ea-
sing o de . We w i e k·k2 o he Euclidean no m on Cn, de ined by kxk2:=
(
P
n
i=1 |xi|2)1/2, and k·k o he associa ed ope a o no m on Cn×n, de ined by
kMk:= sup{kMxk2:kxk2= 1}. We w i e GLn(C) o he g oup o in e ible
ma ices o Cn×n. Gi en ε≥0, he o dina y ε-pseudospec um o Mcan be
de ined as he se Λε(M) := {z∈C:σn(zIn−M)≤ε}.
In he Ph.D. Thesis o M. Ka ow [4] he ela ionship was shown be ween
he condi ion numbe s o eigen alues o a ma ix M∈Cn×n, whose spec um
is {λ1, . . . , λp}, and i s Jo dan decomposi ion
M=
p
X
i=1
(λiPi+Ni),
whe e Piis he Riesz p ojec o co esponding o λiand Niis he eigennilpo en
ma ix associa ed wi h λi. In pa icula , he index ν(λi) o each eigen alue λi
plays a majo ole. Mo eo e , in he same disse a ion, he condi ion numbe o
he eigen alue λiis ela ed o he connec ed componen o he pseudospec um
∗This wo k was suppo ed by he Minis y o Educa ion and Science, P ojec MTM 2007-
67812-CO2-01.
†Depa men o Ma hema ical Enginee ing and Compu e Science, The Public Uni e si y
o Na a e, Campus de A osad´ıa, 31006 Pamplona, Spain. [email p o ec ed]
‡Depa men o Applied Ma hema ics and S a is ics, The Uni e si y o he Basque
Coun y, Facul y o Pha macy, 7 Paseo de la Uni e sidad, 01006 Vi o ia-Gas eiz, Spain,
[email p o ec ed], [email p o ec ed]
1
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in Linea Algeb a and i s
Applica ions 436(6) : 1683-1688 (20212), which has been published in inal o m a h ps://doi.o g/10.1016/
j.laa.2011.01.014. © 2011 Else ie unde CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-
nd/4.0/)
Λε(M) con aining λi. These ac s led us o hink ha he e should be a close
ela ionship be ween he Jo dan canonical o m o Aand i s pseudospec a.
Le kbe an in ege , 1 ≤k≤n. Fo ε≥0, he geome ic ε-pseudospec um
o Mo o de kcan be de ined as he se
Λ(g)
ε,k(M) = {z∈C:σn−k+1(zIn−M)≤ε}.
In his pape we a e going o es ablish ha he geome ic pseudospec a Λ(g)
ε,k(A)
o small enough εde e mine he Jo dan canonical o m o A, o equi alen ly,
de e mine i s in a ian ac o s. This is he con en o he main heo em in his
pape , which is he ollowing.
Theo em 1 (Su icien condi ion o simila i y).Le A, B ∈Cn×n. Le us
assume ha o each z∈C he singula alues o zIn−Aa e he same as hose
o zIn−B. Then Aand Ba e simila ma ices.
This Theo em will be p o ed in Sec ion 3.
Rema k 1. No ice ha i Aand Ba e simila and bo h ma ices a e no mal,
hen o each z∈C he singula alues o zIn−Aa e he same as hose o
zIn−B. This is no longe ue i Aand Ba e no assumed no mal.
Theo em 1 was also inspi ed by Fac 5(b), page 16-2 in Chap e 16 on
Pseudospec a w i en by M. Emb ee in he Handbook o Linea Algeb a, edi ed
by L. Hogben [3]. This Fac says ha i Aand Ba e n×ncomplex ma ices
ha ha e he same o dina y ε-pseudospec um o e e y ε > 0, hen Aand B
ha e he same minimal polynomial. We ema k ha Λε(M) = Λ(g)
ε,1(M).
Once we had p o en ou heo ems, we ead he pape by M. Fo ie Bou que
and T. Rans o d [1], which came o con i m ou hunch. Two ma ices A, B ∈
Cn×na e said o be uni a ily simila i he e exis s a uni a y ma ix U∈Cn×n
such ha B=U∗AU, whe e ∗s ands o he conjuga e anspose. M. F.
Bou que and T. Rans o d say ha he complex n×nma ices Aand Bha e
supe -iden ical pseudospec a i , o each z∈C, he singula alues o zIn−A
a e he same as hose o zIn−B. In [1] i was also p o ed ha his condi ion
is excessi e, and i is su icien o equi e hese equali ies o a ce ain ini e se
Fo C; namely,
Theo em 2. Le F:= { peiθq:p, q = 0, . . . , n}, whe e 0< θ0<· · · < θn< π
and 0< 0<· · · < n. Suppose ha A, B ∈Cn×nsa is y
σk(zIn−A) = σk(zIn−B) (z∈F, k = 1, . . . , n).
Then Aand Bha e supe -iden ical pseudospec a.
Also hey showed ha : (a) A, B ∈C2×2ha e supe -iden ical pseudospec a i
and only i Ais uni a ily simila o B; (b) A, B ∈C3×3ha e supe -iden ical
pseudospec a i and only i Ais uni a ily simila o Bo o i s anspose; (c)
he e exis A, B ∈C4×4wi h supe -iden ical pseudospec a such ha kA2k 6=
kB2k, his implies ha Ais no uni a ily simila ei he o Bo o i s anspose.
We would no e ha he e a e p oblems in pu e ma hema ics and con ol
heo y whe e he simul aneous conside a ion o all he singula alues leads
o mo e sa is ac o y solu ions, like he p oblem o s udying he app oxima ion
2
o a bounded ma ix unc ion on he uni ci cle by bounded analy ic ma ix
unc ions on he uni disc [5].
The o ganiza ion o his pape is as ollows: Gi en M∈Cn×nand z0an
eigen alue o M, we will analyze he asymp o ic beha io o he singula alues
o he cha ac e is ic ma ix zIn−Mwhen z→z0in Sec ion 2. We will p o e
Theo em 1 in Sec ion 3. In Sec ion 4 we will gi e an ex ension o Theo em 1,
and we will ame hese esul s in he heo y o pseudospec a.
2 O de s o he singula alues o a cha ac e is-
ic ma ix as in ini esimals
Le a ma ix M∈Cn×nand z0an eigen alue o M. In his sec ion we will
s udy he asymp o ic beha io o he singula alues o he cha ac e is ic ma ix
zIn−M, when z→z0. To ha end, we need he ollowing no a ions. Le V0(z0)
be a punc u ed neighbo hood o z0in C, we conside he se Fo eal unc ions
de ined on V0(z0). Then, we ha e he ollowing de ini ion.
De ini ion 1. Le , g ∈F. I he e a e cons an s δ, ∆, d > 0 such ha o
e e y z∈B0(z0, d) (open punc u ed disk cen e ed a z0and adius d)
(z)>0, g(z)>0 and δ≤ (z)
g(z)≤∆,
we w i e (wi h Ha dy’s no a ion [2])
(z)g(z) (when z→z0).
We say ha a unc ion ∈Fis an in ini esimal as z→z0i limz→z0 (z) =
0. I (z) |z−z0|k(wi h kin ege ≥1) we say ha (z) is an in ini esimal
o o de kas z→z0. The ela ion is an equi alence ela ion.
Rema k 2. I j, k a e in ege s ≥0 and
|z−z0|j |z−z0|k(z→z0),
hen j=k.
Rema k 3. Recall ha o posi i e unc ions , g ∈F he ela ion (z)∼g(z)
as z→z0means
lim
z→z0
(z)
g(z)= 1.
I is ob ious ha (z)∼g(z) as z→z0implies (z)g(z) as z→z0.
The main esul o his sec ion is he ollowing lemma.
Lemma 3. I Jk(z0)is he k×kJo dan block wi h eigen alue z0, hen, as
z→z0,
σj
zIk−Jk(z0)
∼
¨
1, j = 1, . . . , k −1,
|z−z0|k, j =k.
3
P oo . Wi hou loss o gene ali y, we may suppose ha z0= 0 and w i e simply
Jk:= Jk(0). Since J∗
kJk= diag(0,1,...,1), i ollows ha he singula alues
o Jka e 1,...,1,0. Hence σj(zIk−Jk)→1 as z→0 o j= 1,2, . . . , k −1.
Also
k
Y
j=1
σj(zIk−Jk)2= de ((zIk−Jk)∗(zIk−Jk)) = |de (zIk−Jk)|2=|z|2k,
whence i ollows ha σk(zIk−Jk)∼ |z|kas z→0.
2
Fo he p oo o Lemma 7, we need some p elimina y esul s. The i s one
can be seen in [6].
Lemma 4. Le M1, M2, M3∈Cn×n. Then, o k= 1,2, . . . , n,
σn(M1)σk(M2)σn(M3)≤σk(M1M2M3)≤ kM1kkM3kσk(M2).
Wi h his esul we can p o e he ollowing.
Lemma 5. Le M∈Cn×n, P ∈GLn(C)and z0∈C. Then, o j= 1,2, . . . , n,
σj(zIn−P−1MP)σj(zIn−M) (z→z0).
Lemma 6. Le L∈Cq×qand z0be a complex numbe such ha z0/∈Λ(L).
Then, o j= 1,2, . . . , q,
σj(zIq−L)1 (z→z0).
P oo . Fo j= 1,2, . . . , q, he limi
lim
z→z0
σj(zIq−L) = σj(z0Iq−L)
is nonze o and ini e.
2
Lemma 7. Le Jbe he Jo dan o m o a ma ix M∈Cn×n. Le z0∈Cand
k∈ {1, . . . , n}. Then he numbe o k×kJo dan blocks in Jwi h eigen alue z0
is equal o he numbe o j∈ {1, . . . , n}such ha σj(zIn−M) |z−z0|kas
z→z0.
P oo . By Lemma 6 i z0/∈Λ(M), hen o j∈ {1, . . . , n},
σj(zIn−M)1 (z→z0);
so, in his case, he e is no jsuch ha σj(zIn−M) |z−z0|kas z→z0.
I z0∈Λ(M), by Lemma 5, o j= 1, . . . , n,
σj(zIn−M)σj(zIn−J) (z→z0).
Le
J=J0⊕J1,
4
whe e J0∈Cn0×n0is he di ec sum o he Jo dan blocks associa ed wi h z0,
and z0/∈Λ(J1). When zis su icien ly close o z0, he las singula alues o
zIn−Ja e jus he in ini esimal singula alues o zIn0−J0as z→z0. Thus,
lim
z→z0
σj(zIn−J)=0, o j=n− + 1, . . . , n −1, n.
The numbe o j∈ {n− +1, . . . , n−1, n}such ha he o de o he in ini esimal
σj(zIn−J) as z→z0is k, is equal o he numbe o k×kJo dan blocks in J0
associa ed wi h z0. Fo j∈ {1, . . . , n − }, we ha e σj(zIn−J)1 as z→z0.
2
3 P oo o he main esul
In his sec ion, we will p o e he main esul o his pape .
P oo o Theo em 1.
Le M∈Cn×nand z0∈C. Then z0∈Λ(M) i and only i σn(z0In−M) = 0.
Since o each z∈C,σn(zIn−A) = σn(zIn−B), he eigen alues o Aand B
a e he same,
Λ(A) = Λ(B) = {λ1, λ2, . . . , λp}.
As σj(zIn−A) = σj(zIn−B) o z∈Cand j∈ {1, . . . , n}, hen o each
k∈ {1, . . . , n}and λi∈Λ(A), he numbe o j∈ {1, . . . , n}such ha
σj(zIn−A) |z−λi|kas z→λi
is equal o he numbe o j∈ {1, . . . , n}such ha
σj(zIn−B) |z−λi|kas z→λi.
Thus, by Lemma 7, he numbe o k×kJo dan blocks associa ed wi h λiin
he Jo dan o ms o Aand Bis he same. Gi en ha his holds o e e y
λi∈Λ(A) = Λ(B), we in e ha Aand Ba e simila .
2
4 Rema ks
Following a line o easoning simila o ha o Theo em 1, we can es ablish he
ollowing heo em.
Theo em 8. Le A∈Cn×n, B ∈Cm×m. Le us suppose ha n≥mand le
gi(λ)|gi+1(λ)|···|gm−1(λ)|gm(λ)
be he non i ial in a ian ac o s o B. Le us assume ha o each z∈Cand
k= 1,2, . . . , m −i+ 1,
σn−k+1(zIn−A) = σm−k+1(zIm−B) (1)
Then he las m−i+ 1 in a ian ac o s o A,
n−m+i(λ)| n−m+i+1(λ)|···| n−1(λ)| n(λ),
5
a e non i ial, and
n(λ) = gm(λ), n−1(λ) = gm−1(λ), . . . , n−m+i(λ) = gi(λ).
Le M∈Cn×n. Fo e e y eal numbe ε≥0, ano he equi alen de ini ion
o he o dina y ε-pseudospec um o Mis
Λε(M) :=
[
X∈Cn×n
kX−Mk≤ε
Λ(X).
Fo z∈Cwe deno e by gm(z, M) he geome ic mul iplici y o zas eigen alue
o M. I z /∈Λ(M), we ag ee ha gm(z, M) = 0. Le kbe an in ege , 1 ≤k≤n,
and le Λ(g)
k(M) deno e he se o z∈Λ(M) such ha gm(z, M)≥k. Fo ε≥0,
he geome ic ε-pseudospec um o Mo o de kcan be de ined, al e na i ely,
by
Λ(g)
ε,k(M) :=
[
X∈Cn×n
kX−Mk≤ε
Λ(g)
k(X).
5 Conclusions
Le A, B be n×ncomplex ma ices such ha he singula alues o zIn−Aa e
he same as hose o zIn−B o each z∈C. Then Aand Ba e simila . A mo e
gene al esul o squa e ma ices Aand Bo dis inc sizes has been s a ed.
Acknowledgemen
We hank he e e ee o he c i icisms, sugges ions and he p oo o Lemma 3.
Re e ences
[1] M. Fo ie Bou que, T. Rans o d. Supe -iden ical pseudospec a, J. Lon-
don Ma h. Soc. (2) 79 (2009) 511–528.
[2] G.H. Ha dy. O de s o in ini y. Ha ne Publishing Company, New Yo k,
1971.
[3] L. Hogben. Handbook o Linea Algeb a. Chapman, Hall/CRC, 2007.
[4] M. Ka ow. Geome y o spec al alue se s. Ph.D. Thesis. Uni e si y o
B emen, 2003.
[5] V.V. Pelle , N.J. Young. Supe op imal analy ic app oxima ions o ma ix
unc ions. J. Func . Anal. 120 (1994) 300–343.
[6] J. F. Quei ´o, E. Ma ques de S´a. Singula alues and in a ian ac o s o
ma ix sums and p oduc s. Linea Algeb a Appl. 225 (1995)43–56.
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