Pub
.
Ma
.
UAB
N° 22
No
.
1980
Ac es
VII
JMHL
MATRICIALNORMS
:
SOME
APPLICATIONS
AND
OPEN
QUESTIONS
José
Vi ó ia
Dp o
.
de
Ma ema£ica
.
Uni e sidade
de
Coimb a
Abs ac
-
We
men íon
some
heo e ical
aspec s
o
ma icialno ms,
including
some ecen
de elopmen s
E
obe , Deu sch, Baue ,
Ba ke ,
Meixne ,
Coimb a=
.
Then
we
e e
se e alapplica ions
o
ma icial
no ms
:
con e gence
o
i e a i e
p ocedu es,
weak
con ac ion
in
ec o ial
no m/ ixedpoin
ques ions
E
obe l
;
app oxima ion
in
-me ic
spa
ces
ICoimb a
:
;
localiza ion
o
ze os
o
polynomials
Eeu sch,
Vi-
ó ial
;
bounds
o
la en
oo s
o
lambda-ma ices
E
i ó ia
:
;
i-
b a ing
sys ems
c
limaco}Vi ó ia
:
;
lowe
bounds
o
linea
mappings
E
ode=
.
Finally, we
include
some open
ques ions
conce ning
bo h
heo e ical
and p ac ical
aspec s
.
1
.
Applica ions
o
ma icial
nonns
In
his
sec ion
we
e e
some
applica ions
o
p ope ies
o
ec o ial
Ea icial
:
no ms
.
1
.1
.
Con e gence
o linea
i e a ions
One
u ilizes
p ope ies
o
ma icial
no ms
o
con ol
he
con e gence
o
i e a i e
p ocedu es-speed
and
e o
.
.
Block-i e a ions
:
Gauss-Ssidel,
Ja-
cobi,
o e - elaxa ion
.
*
The
wo k
o
his
pape
was
suppo ed,
a
di e en
s ages,
by
Fundagáo
Calous e
Gulbenkian
(Lisboa,
Po ugal),
Uni e sidade
de
Mapu o
(Mozambique)
and Ins i u o
Nacional
de
In es iga gáo
Cien í ica
(Lisboa,
Po ugal)
.
1
.2
.
"Sé ie-pa allee"
i e a i e
me hods
.
One
cons uc s
algo i hms
o linea
i e a ions
i ed
o
au oma ic
calcula ions
"in
pa allel"
-
se e al
a i hme ic
uni s,
wo king
simul aneously,
b anched
o a
common
memmo y
.
These
algo i hms
can
be
placed a
an
in e me
dia y
le el
be ween
he
Gauss-Seidelme hod
-
sequen ial
-
and
he Jacobime-
hod
-
pa allel
.
1 .3
.
Weakcon ac ion
in
ec o ial
noim/ ixed
poin
p oblems
Pe on-F obenius
heo y
o
ma ices
pa i ioned
in o
blocks
.
1 .4
.
Con e gence
o
ma ix
sequences
The e
a e
esul s,
using
(scala )
no ms, on
con e gence
o
ma ix
se
quences
.
Such
esul s
can
be
gene alized
by
using
ec o ial
no ms
.
1
.5
.
Bounds o he
ze os
o
polyncmials
One
akes
he
companionma ix

C

o
a
polynomial

p(z)
=a
o
zn
+
a1zn-1
+
+
. .
.+a
n-l
z+
an
,a
iE
C,
(i=0,1,
.
.
.,
n)

and
cons uc s
se e al
adequa ed
ma i-
cial
no ms
ela ed
o
ma ix
C
.
So we
ge
uppe
bounds o
he
spec al
a-
dius
o
ma ix
C,
i
.e
.
o
he
absolu e
alues o
he
oo s
z
o p(z)
.
1
.6
.
Bounds o
he
la en
oo s
o
lambda-ma ices
-

-

We
conside
he
lambda-ma ix
M(X)
=Ion+A1
an-1+
..
.+An-l
a+An,
whe e
Ai
,
I
EM(p
C)
p
,
(i=l,
. .
. .
n)
.

We

need
o

know
he

ze os
o
de
M
(a)
=
I
IX
n+Al
an-!+
.
. .
.
..
+An-l
a+A
n
1
=0,

ha
is
o
soy, we need o know
he la en
oo s

a

o
he
lambda-ma ix
.
M(a)
.
The
calcula ion
o
he
la en
oo s
may become
a
o mi-
dable
ask,
his
depending
on
n
and
p
.
Ins ead
o
calcula ing
hem,
we look
-
Po
localiza ion
domains
o
he
la en
oo s
.
We
associa e
o
he
lambda-ma ix
i s
block-companion
ma ix
M
.
The
eigen alues
o
block-companion
ma ix
M
a e
he
la en
oo s
o
he
lambda-
-ma ix
.
Then
we
cons uc
se e al
ma icial
no ms
ela ed
o
he
ma ix
M,
and ge
uppe
bounds
o
IXI
.
Inciden ally,
we
ob ain
known
(and new)
esul s
o
ze os
o
(scala )
polynomials
.
1 .7
.
App oxima ion
in
-me ic
spaces
Gene aliza ions
o
-me ic
spaces
i
seudo-me ic
spaces
:
o
esul s
in
me ic
spaces
.
Gene aliza ion
o
New on's
me hod
o
supe
-
-
me ic
spaces,
he
ec o ialme ic
being
induced
by a
ec o ial
no m
.
1 .8
.
Lowe
bounds
o
linea
mappings
Gi en
a
( egula )
ec o ial
no m
T
o
o de
k
un
k
n
.
Lowe bound
o
a
ma ix
Co
:pe a o ]
A
eM
n
~
n
(K)

wi h
espec
o
he
ec o ial
no m
T is
n
a
ma ix
NEMk,k(R)

such
ha

VxeK
,
N
4'
(x)
= (Ax)
.
We
can
apply hese
lowe
bounds
o
he
es ima ion
(in
ec o ial
no m)
o , he
ob ained
app oxima ion
o
he
solu ion
o
a
linea
sys em
.
blocks
.
1
.9
.
Eigen alues
in
block-pa i ioned
ma ices
Gene aliza ion
o
Ge schgo in
esul s
o
ma ices
pa ioned
in o
1
.10
.
Vib a ing
sys ems
Fo
s udying
he
mo ion
equa ions,
one
can
use
(classical)
Lag ange
me hod
.
The
solu ion
o
such
equa ions
will
ep esen
a
sinusoidal
mo ion,
so
we
lock
o solu ions
o
an
equa ion
o ype
A~+Bp+Cp
=
P
.
Fo
sol ing
hese
equa ions
we
need
o
know
he
la en
oo s
o
he
lambda-ma ix

M()=AX
2
+
+Ba+C
.
Bu
in
ce ain
si ua ions
we do
no
need
o
calcula e
he
la en
oo s
o
lambda-ma ix
M(X)
.
I
su ices
o
know
a
localiza ion
domain
o
he
la-
en
oo s
.
In his case
a e
well sui ed
he
esul s
e e ed
o
in
1
.6
.
2
.
Open
ques ions
In
his
sec ion
we
men ion
some
seemingly
open
ques ions,
in ol ing
ma icial
no ms
:
2
.1
.
De ia ion
o
a
ma ix
omsingula i y
2 .2
.
De ia ion
o
a
ma ix
om
noimali y
2
.3
.
Spec al
a ia ion
o
wo
ma ices
2
.4
.
Gene aliza ions
o
a
undamen al
inequali y
un
ma icial
no ms
(*)

p
(A)
<p
[,D(A)7

,

p
(M)

:

Spec al
adius o
a
ma ix

Mi
E
M)

:
ma icial
no m
o
a
ma ix

M
.
We
aise
he
ollowing
ques ions
:
a) I is
possible
o
conside
a
ela ion
"simila "
o (*) in
he
con ex
o
ma ices
whose
elemen s
a e
in e als?
b)
How
"seems"
ela ion
(*)
i
he
elemen s
o
ma ix
A
be-
long
o
a
p .i .
.
(p incipalideal
aing)?
u1
."2
.
.
. .
."k
ha
c)
Le

X
lI
x2

..
.x n

be
he
eigen alues
o

A
EM
n
.n
(K)

and
be
he
eigen alues
o
i s ma icial
no m
(D(A)
CMk
k
(R)
.
Supp9se
1
x
111
1X21
>
.
-_
I
X,
I

and

l"11
1I"21
1
.
. .
2
!"
k
l
F om
(*)
we
ha e
Ixll
=
P(A) <P
(D(A))
=
lull
Wha
can
we
say
abou
ma ices
such ha
I
x2
1_
I"
2
1

1x
3
1=1"
3
1

.
Rema k
A
comple e sion
o
his
pape
will
appea
elsewhe e,
PRINCIPAL
REFERENCES
e c
. ?
A
.
BODE

-
Symme ien
und
An isymme ien
de
Robe 'schenQuasio dnung
in
Ma izen aumen,
Lin
.
Alg
.
and
Appl
.
23
:87-99 (1979)
.
J
.
CLIMACOj
J
.
VITóRIA
-
Sis emas
Vib a ó ios,
ma izes-lambda
e
no mas
ec-
o iais
.
I
Encon o
de
Físicos
e
Ma emá icos,
Coimb a,
11-12
Ou ub o,
1979
.
E
.
COIMBRA
-
Ap oximagóes
em
espagos -mé icos
.
Tese
.
Uni e sidade
No a
de
Lisboa,
Lisboa,
1979
.
F
.
ROBERT

-
Ma ices
non-nága i es
e
no mes
ec o ielles
.
(Cou s
O .E
.A .)
.
Uni
.
G enoble
.
1974
.
J
.
VITóRIA
-
No mas
ec o iais
de
ec o es
e de
ma izes
.
Repo
.
Uni e si
dade
de
L
.Ma ques (now
Uni e sidade
de
Mapu o)
.Mozambique
.
1974
.
Ma icial
no ms
and
he
oo s
o
polynomials
.
Re ,
Cienc,
Ma
.
5
:
51-56
(1974-75)
.
Ma icial
no ms
and
lambda-ma ices
.
Re
.Cien
.Ma
.
5
:
11-30
(1974-75)
.
Gene aliza ion
o
inequali ies_by
Ca michael
and
Mason
and
by
Pa odi
.
Compu ing,
22
:
363-365
(1979)
.
Ma icial
no ms
and
he
di e ences
be ween
he
ze os
o
de-
e minan s
wi h,polynomial
elemen s
.
Lín
.
Alg
.
and
¡ e
Appl
.
28
:
279-283
(1979)
.
La en
oo s
o
lambda-ma ices,
K onecke
sums
and
ma icial
no ms
.
Submi ed
.
Uppe
bounds
o he
la en
oo s
o
lambda-ma ices
.
Submi ed
.