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The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society.
Volume 6, pp. 20-30, March 2000.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela

ELA

ON THE RELATION BETWEEN THE NUMERICAL RANGE
AND THE JOINT NUMERICAL RANGE



OF MATRIX POLYNOMIALS

P. J. PSARRAKOSy

AND

M. J. TSATSOMEROSz

It is shown that the numeri

al range, NR[P ()℄, of a matrix polynomial P () =
onsists of the roots of all s
alar polynomials whose
oeÆ
ients
orrespond
to the elements of the
onvex hull of the joint numeri
al range of the (m +1)-tuple (A0 ; A1 ; : : : ; Am ).
Moreover, the elements of the joint numeri
al range that give rise to s
alar polynomials with a
ommon root belonging to NR[P ()℄ form a
onne
ted set. The latter fa
t is used to examine the
multipli
ity of roots belonging to the interse
tion of the root zones of NR[P ()℄. Also an approximation s
heme for NR[P ()℄ is proposed, in terms of numeri

al ranges of diagonal matrix polynomials.
Key words. joint numeri
al range, matrix polynomial, root zone
AMS subje
t
lassi
ations. 15A60, 47A12
Abstra
t.

Am m + : : : + A1  + A0

1. Introdu
tion. The numeri
al range of a matrix polynomial and the joint
numeri
al range of a set of matri
es (usually taken to be the
oeÆ
ients of a matrix

polynomial) are both notions that generalize the
lassi
al numeri
al range of a matrix.
The former is a one-dimensional generalization and the latter a multi-dimensional
generalization. We will
ontinue previous eorts found in our referen
es to study the
relations between these sets and dis
over their properties. The
on
luding paragraph
of this se
tion in
ludes a brief des
ription of our results, requiring the following basi
terminology and notation.
Let Mn (C ) denote the algebra of n  n matri
es over the
omplex eld and let

(1)

P () = Am m + : : : + A1  + A0

be a matrix polynomial, where Aj 2 Mn (C ) (j = 0; 1; : : : ; m), Am 6= 0. The positive
integers m and n are referred to as the degree and the order of P (), respe
tively.
The s
alar 0 2 C is
alled an eigenvalue of P () if the equation P (0 )x = 0 has a
nonzero solution x = x0 2 C n . Thus the spe
trum of P () is the set
 ()℄ = f 2 C : det P () = 0g:
As det P () is a polynomial of degree at most n
m,  ()℄ is either equal to C or
it
onsists of at most n
m
omplex numbers. The numeri
al range of P () is the set

NR[P ()℄ =

 2 C : x P ()x = 0 for some nonzero x 2 C n g;

f

 Re
eived by the editors on 6 May 1999. A

epted for publi
ation on 27 February 2000. Handling
editor: Bryan L. Shader
y Department of Mathemati
s and Statisti
s, University of Regina, Regina, Saskat
hewan, Canada
S4Sz 0A2 (panosmath.uregina.
a).
Department of Mathemati
s and Statisti

s, University of Regina, Regina, Saskat
hewan, Canada
S4S 0A2 (tsatmath.uregina.
a). Resear
h supported by NSERC resear
h grant.
20

ELA
Joint Numeri
al Range

21
[P ()℄ generalizes

whi
h is
losed and
ontains  ()℄. Noti
e that the notion of

the
lassi
al numeri
al range of a matrix A, that is, the
onvex set
[I A℄ = fxAx : x 2 C n; xx = 1g:
Also noti
e that 0 2 [P ()℄ if and only if 0 2 [I P (0)℄. It is known
that [P ()℄ is not ne
essarily
onne
ted and that it is bounded if and only if
0 62 [I Am℄; see Li and Rodman [6℄.
The joint numeri
al range of P () is the set of
omplex (m + 1)-tuples
[P ()℄ = f(x A0x; x A1 x; : : : ; xAmx) : x 2 C n; xx = 1g:
The joint numeri
al range, being a
ontinuous image of the unit sphere, is

ompa
t
and
onne
ted but not ne
essarily
onvex; see Binding and Li [1℄. Its
onvex hull is
denoted by Cof [P ()℄g. Observe that
[P ()℄ = f 2 C :
mm + : : : +
1 +
0 = 0; (
0 ;
1; : : : ;
m) 2 [P ()℄g:
Consider now the following two subsets of C : the set of roots of all m-degree
s
alar polynomials whose
oeÆ

ients
orrespond to elements of the joint numeri
al
range of m +1 given matri
es, and the set of roots of all m-degree s
alar polynomials
whose
oeÆ
ients
orrespond to elements of the
onvex hull of the joint numeri
al
range of the same m + 1 matri
es. As we will see, both these sets
oin
ide and are
equal to the numeri
al range of the asso
iated matrix polynomial, despite the fa
t

that the joint numeri
al range is not ne
essarily
onvex (Proposition 2.1). Moreover,
the elements of the joint numeri
al range asso
iated with s
alar polynomials that
have a
ommon root belonging to the numeri
al range, form a set that is not only
nonempty but also
onne
ted (Proposition 2.2). Whether or not the multipli
ities of
roots of s
alar polynomials with
oeÆ
ients from the two sets above also
arry over

is an open (and seemingly substantial) problem with impli
ations in the fa
torization
of matrix polynomials. Some partial results in this regard
an be found in se
tion 3.
Finally, in se
tion 4 we des
ribe a pro
ess to approximate the numeri
al range of a
matrix polynomial using the (joint) numeri
al ranges of diagonal matri
es.
Noti
e that if Q() is a
matrix polynomial of degree m, then
[P ()℄  [Q()℄ =) [P ()℄  [Q()℄:
More generally, given a set G of
omplex (m + 1)-tuples su
h that
[P ()℄  G,
(2)
[P ()℄  f 2 C : bmm + : : : + b1 + b0 = 0; (b0; b1; : : : ; bm) 2 Gg:
The following strengthening of (2) holds when G = Cof [P ()℄g.
For every matrix polynomial P () of degree m we have
NR[P ()℄ = f 2 C :
m m + : : : +
1  +
0 = 0; (
0 ;
1 ; : : : ;
m ) 2 CofJNR[P ()℄gg:
NR

NR

NR

NR

NR

NR

JNR

JNR

NR

JNR

2. The
onvex hull of the joint numeri
al range.

JNR

JNR

NR

NR

JNR

NR

JNR

Proposition 2.1.

ELA
22

P. J. Psarrakos and M. J. Tsatsomeros

Proof. Let 0 2 C su
h that
m m
0 + : : : +
1 0 +
0 = 0

(3)

for some
= (
0 ;
1 ; : : : ;
m ) 2 CofJNR[P ()℄g. By Caratheodory's Theorem applied
to C m+1 ,
is a
onvex
ombination of   2(m + 1) + 1 points q1 ; q2 ; : : : ; q 2
JNR[P ()℄, that is,
=


X
j =1

tj qj ;


X
j =1

Sin
e there exist unit ve
tors xj

tj = 1; tj

0

(j = 1; 2; : : : ; ):

2 C n su
h that

qj = (xj A0 xj ; xj A1 xj ; : : : ; xj Am xj ) (j = 1; 2; : : : ; );

we have that

k =


X
j =1

tj (xj Ak xj ) (k = 1; 2; : : : ; m)

and thus (3) be
omes

X
j =1






tj (xj Am xj )m
0 + : : : + (xj A1 xj )0 + (xj A0 xj ) =


X
j =1




tj xj P (0 )xj = 0:

Sin
e all the
omplex numbers xj P (0 )xj (j = 1; 2; : : : ; ) belong to the
onvex set
NR[I P (0 )℄, it follows that 0 2 NR[I P (0 )℄, i.e., 0 2 NR[P ()℄. Hen
e

f 2 C

:
m m + : : : +
1  +
0 = 0; (
0 ;
1 ; : : : ;
m ) 2 CofJNR[P ()℄g  NR[P ()℄:

The opposite in
lusion follows from the dis
ussion pre
eding this proposition.
We
ontinue with an interesting geometri
feature of the relation between
NR[P ()℄ and JNR[P ()℄. Suppose that
m m
0 + : : : +
1 0 +
0 = 0. Then there
exist a0 ; a1 ; : : : ; am 1 2 C su
h that

m m + : : : +
1  +
0 = ( 0 )(am 1 m 1 + : : : + a1  + a0 )
= am 1 m + (am 2 0 am 1 )m 1 + : : : + (a0

0 a1 )

0 a0 :

We may then dene the set of
omplex (m + 1)-tuples
(4) T1 (0 ) = f(
0 ;
1 ; : : : ;
m ) :
m m
0 + : : : +
1 0 +
0 = 0g
= f( 0 a0 ; a0 0 a1 ; : : : ; am 1 ) : (a0 ; a1 ; : : : ; am 1 ) 2 C m g
= spanf( 0 ; 1; 0; : : : ; 0); (0; 0 ; 1; 0; : : : ; 0); : : : ; (0; : : : ; 0; 0 ; 1)g:
By Proposition 2.1, if T1 (0 ) interse
ts CofJNR[P ()℄g, it also interse
ts JNR[P ()℄.
However, the following more general result holds.

ELA
Joint Numeri
al Range
Proposition 2.2.
polynomial with

n

3

.

Let

23

= Am m + : : : + A1  + A0 be an n  n matrix
every ! 2 NR[P ()℄ the set T1 (! ) \ JNR[P ()℄ is

P ()

For

onne
ted.

Proof. The set T1 (! ) \ JNR[P ()℄
onsists of all points (
0 ;
1 ; : : : ;
m ) su
h that
m ! m + : : : +
1 ! +
0 = 0 and (
0 ;
1 ; : : : ;
m ) = (x0 A0 x0 ; x0 A1 x0 ; : : : ; x0 Am x0 ) for
some x0 2 C n . It is shown in Lyubi
h and Markus [8℄ that the set

L0 = fx 2 C n : x x = 1 and x P (! )x = 0g

is
onne
ted for n  3. Consequently, if we
onsider the
ontinuous fun
tion
 : fx 2 C n : x x = 1g ! JNR[P ()℄
dened by (x) = (x A0 x; x A1 x; : : : ; x Am x), then (L0 ) = T1 (!) \ JNR[P ()℄ is
indeed a
onne
ted set.
In the spe
ial
ase when P () is self-adjoint, that is, the
oeÆ
ient matri
es
are Hermitian, JNR[P ()℄
onsists of real (m + 1)-tuples. Therefore for every ! 2
NR[P ()℄ \ R we
an dene the real hyperplane
= f( !a0; : : : ; am 2 !am 1 ; am 1 ) : (a0 ; a1 ; : : : ; am 1 ) 2 R m g
= spanf( !; 1; 0; : : : ; 0); (0; !; 1; 0; : : : ; 0); : : : ; (0; : : : ; 0; !; 1)g:
Similarly to Proposition 2.2, it
an be shown that if P () is an nn matrix polynomial
with n  3, then T1 (!)R \ JNR[P ()℄ is
onne
ted for every ! 2 NR[P ()℄ \ R .
3. Root zones of the numeri
al range. Let P () be a matrix polynomial
as in (1), whose numeri
al range is bounded, that is, 0 62 NR[I Am ℄. We
an always number the roots of x P ()x = 0 to obtain a sequen
e of fun
tionals
1 (x); 2 (x); : : : ; m (x) on the unit sphere S , having ranges 1 ; 2 ; : : : ; m , respe
tively. We willSrefer to 1 ; 2; : : : ; m as root zones of NR[P ()℄. We
learly have that
m
NR[P ()℄ =
j =1 j ; see Isaev [5℄. Lyubi
h [7℄ refers to the j as `root bran
hes' and
uses them in identifying
onditions that are equivalent to the roots of x P ()x = 0
being simple at every point of S .
If there exists a nonzero x 2 C n su
h that x P ()x = 0 has multiple roots,
then some root zones have nonempty interse
tion and thus NR[P ()℄ has fewer than
m
onne
ted
omponents. In what follows, we will
onsider the
onverse, namely,
whether the nonempty interse
tion of some of the root zones ne
essarily implies the
existen
e of multiple roots; see Maroulas and Psarrakos [9, Theorem 3.1℄ for the selfadjoint
ase. For that purpose, we will use the notion of the joint numeri
al range
and the results of the previous se
tion. We will also need to restri
t our attention to
a
ompa
t subset of S , where the fun
tionals 1 (x); 2 (x); : : : ; m (x) are well dened
and
ontinuous. We expand on this requirement next.
Certainly, the roots of x P ()x = 0 depend
ontinuously on the
oeÆ
ients

x Aj x, and thus on x. This fa
t
an be dedu
ed from Ostrowski [10, pp. 334{335℄ by
onsidering the
ompanion matrix of the polynomial
m + bm 1 m 1 + : : : + b1  + b0 ;
T1 (! )R

ELA
24

P. J. Psarrakos and M. J. Tsatsomeros

where
bj

x A x

=  j (j = 0; 1; : : : ; m
x Am x

1):

In spite of the
ontinuous dependen
e of the roots of x P ()x = 0 on x, it may not be
possible to dene the aforementioned fun
tionals 1 (x); 2 (x); : : : ; m (x)
ontinuously
on the whole unit sphere S . This situation is illustrated by the following example.
Consider the matrix polynomial
P ()

= I 2



0 2
0 2



and let x = (
os e i  ; sin )T parameterize S . We then have that
x P ()x

= 2

sin(2)ei :

Thus, if we require
ontinuity of the roots 1 (x); 2 (x) as fun
tions of x, then it is
easy to verify that they
an not be dened globally on S .
When the roots of x P ()x = 0 are simple for every x 2 S , a unique global
denition of ea
h j (x) as a
ontinuous fun
tional on S is possible and is indeed
obtained in Lyubi
h [7℄. Otherwise, to pro
eed with our goal, we need to modify the
notion of a root bran
h j (x), using in essen
e the arguments in [7, pp. 55{56℄. We
will dene
ontinuous fun
tionals j = j (x) on a
ertain
ompa
t subset M of S ,
using notions from dierential geometry for whi
h our general referen
e is Helgason
[4℄. We
onstru
t M under the assumption that the polynomial
L(; x)

 x P ()x 2 C [; x1 ; x2 ; : : : ; xn ℄

has no multiple irredu
ible fa
tors. Let D(x) be the dis
riminant of L(; x) with
respe
t to the indeterminate , that is, D(x) is the resultant of the polynomial L(; x)
 L(; x). If D(x)  0 in the polynomial ring C [; x ; x : : : ; x ℄,
and its derivative 
1 2
n
then L(; x) has a repeated fa
tor not in C [x℄;
f. Walker [11, p. 25℄). Therefore,
under our assumption, D(x) is a non-zero element of C [x℄.
Consider now the set
U

= [ C n n f0g ℄ n fx 2 C n : L(; x) =
= fx 2

Cn

n f0g : D(x) 6= 0g:


L(; x)


= 0 for some  2 C g

The set U is open and is the
omplement of an algebrai
hypersurfa
e in C n . A
omplex algebrai
hypersurfa
e has real
odimension 2 in the whole spa
e. Thus by
[4, p. 346, Corollary 12.5℄, U is ar
wise
onne
ted. It is
lear that U is dense in C n .
Therefore the set
U1

= fx 2 C n : x x = 1; D(x) 6= 0g

is an ar
wise
onne
ted dense open subset of the unit sphere S . The set U1 is not
ne
essarily simply
onne
ted. Consider the universal
overing spa
e M0 of U1 . We

ELA
25

Joint Numeri
al Range

an dene on M0 analyti
fun
tionals 1 ; 2 ; : : : ; m being roots of x P ()x = 0, via
a method similar to that mentioned in [7℄. Denote by 0 the
overing map of M0 onto
U1 . For every p 2 M0 , these fun
tionals satisfy i (p) 6= j (p) for 1  i < j  m. For
every pair p; q of points of M0 satisfying 0 (p) = 0 (q), there is a permutation  of
f1; 2; : : : ; mg for whi
h
( ) = (i) (p) (i = 1; 2; : : : ; m):

i q

Dene now an equivalen
e relation  on M0 as follows.
p

 q if and only if

( ) = 0 (q) and

0 p

( ) = i (q) (i = 1; 2; : : : ; m):

i p

Dene now M as the
oset spa
e M0 = . Then M is also a
overing spa
e of U1 .
Let  denote the
overing map of M onto U1 . The fun
tionals j are well dened
on M . For every pair p; q of M for whi
h (p) = (q) and p 6= q, there is an index
1  i  m with i (p) 6= i (q). For every x 2 U1 , the number L of points p 2 M
for whi
h (p) = x is the same and not greater than m!. Thus, M is an analyti
manifold with the natural Riemannian stru
ture. Denote by d the metri
fun
tion on
M or S . As earlier, using the results in [10, pp. 334{335℄, we
an in fa
t dedu
e that
the fun
tionals j are uniformly
ontinuous on M . Taking an arbitrary  > 0, as S is
ompa
t, there are points y1 ; y2; : : : ; y` in S su
h that
S

=

[` f

y

j =1

2 S : d(y; yj ) < =2g:

Sin
e U1 is dense in S , there exist x1 ; : : : ; x` 2 U1 with d(xj ; yj ) < =2 (j = 1; 2; : : : ; `).
Thus
U1

=

[` f

x

j =1

2 U : d(x; xj ) < g:
1

For ea
h j , there exist points pj;1 ; pj;2; : : : ; pj;L of M for whi
h (pj;k ) =
1; 2; : : : ; L) and pj;s 6= pj;t for 1  s < t  L. Then,
M

=

[` [L f

x

j =1 k=1

xj

(k =

2 M : d(x; pj;k ) < g:

Sin
e  > 0 is arbitrary, the spa
e M is totally bounded. The
ompletion M of M
is
ompa
t. By uniform
ontinuity, the fun
tionals j are dened and
ontinuous on
M .
The map 
an be extended to a
ontinuous map of M onto the unit sphere
S . Based on the above dis
ussion and notation, we will in the sequel refer to the
root zones of NR[P ()℄ (where P () is a matrix polynomial as in (1) with bounded
numeri
al range) as the sets
j = fj (x) : x 2 Mg (j = 1; 2; : : : ; m):

ELA
26

P. J. Psarrakos and M. J. Tsatsomeros

Ea
h root zone  is a
ompa
t subset of the Gaussian plane C .
For 0 2 NR[P ()℄
onsider now a
ompa
t set F  M that
ontains a ve
tor x0
su
h that x0 P (0 )x0 = 0. Viewing  (x) as a fun
tion of the
oeÆ
ients of x P ()x,
dene the sets
j

j

Vj;F (0 ) = f(
0 ;
1 ; : : : ;
m ) = (x A0 x; x A1 x; : : : ; x Am x) : x 2 F and
0 = j (x) := j (
0 ;
1 ; : : : ;
m )g (j = 1; 2; : : : ; m)

and
TF (0 ) = f(
0 ;
1 ; : : : ;
m ) = (x A0 x; x A1 x; : : : ; x Am x) : x 2 F g \ T1 (0 );

where T1 (0 ) is as dened in (4). Clearly,
TF (0 )

=

[
m

j

Theorem 3.1.

A1  + A0 .
NR[P ()℄.

=1

Vj;F (0 ):

n  n matrix
NR[I
Am ℄ and

Consider the

Assume that

0 62

If

if the set

F  M
fx 2 S : x P (!)x = 0g \ F

6=i;j

x1 ; x2 2 F

there exist

, 
i

j

= A  + ::: +
m

m

be distin
t root zones of

)
k

k

and for a
ompa
t

let

[

! 2 (i \ j ) n (

P ()

polynomial

su
h that

i (x1 ) = j (x2 ) = ! , and
x0 2 F su
h that

is
onne
ted, then there exists

()x0 = 0.
. Note that sin
e the set fx 2 S :
( )x = 0g \ F is
onne
ted, following the proof of Proposition 2.2, we
an show that T (!) is a
onne
ted subset of
JNR[P ()℄. Moreover,
i (x0 ) = j (x0 ),

namely,

!

x0 P

x P !

is a double root of

Proof

F

TF (! ) = Vi;F (! ) [ Vj;F (! );

where these three sets are
ompa
t subsets of JNR[P ()℄. Consequently, V (!) \
V (! ) 6= ; and there exists x0 2 F su
h that  S
(x0 ) =  (x0 ) = !.
We
omment that when ! 2 ( \  ) n ( 6=  ), then there always exist
p; q 2 M su
h that  (p) =  (q ) = ! . However, for the existen
e of the ve
tor x0 as
in the above theorem, p and q must be
onne
ted by a
ontinuous
urve in
i;F

j;F

i

i

i

j

j

k

i;j

k

j

fx 2 S : x P (!)x = 0g \ M;
a fa
t implied by the remaining assumptions of Theorem 3.1.
Theorem 3.2. Consider the n  n matrix polynomial P () = A  + : : : + A1  +
A0 . Assume that 0 62 NR[I A ℄ and let  ,  be distin
t root zones of NR[P ()℄.
If there exists a
ompa
t F  M and x1 ; x2 2 F su
h that  (x1 ) =  (x2 ) = ! , and

if the set fx 2 S : x P (! )x = 0g \ F is
onne
ted, then there exists x0 2 F su
h that
! is a multiple root of x0 P ()x0 = 0.
m

m

i

m

j

i

j

ELA
27

Joint Numeri
al Range

Proof. As in the proof of Theorem 3.1, the sets
[
Vi;F (! ) and

k

are nonempty and
ompa
t subsets of F

Vk;F (! )

6=i

 M . Moreover, the set

2
[
T (! ) = V (! ) [ 4 V
F

i;F

k

6=i

k;F

3
(! )5

is a
ompa
t and
onne
ted subset of JNR[P ()℄. Hen
e,

2
[
V (! ) \ 4 V
i;F

k

that is, there exists x0

2F

6=i

k;F

3
(! )5 6= ;;

su
h that i (x0 ) = k (x0 ) = ! for at least one k 6= i.

4. Polyhedra and the joint numeri
al range. Consider two matrix polynomials P () = Am m + : : : + A1  + A0 and Q() = Bm m + : : : + B1  + B0 , not
ne
essarily of the same order. If JNR[P ()℄  JNR[Q()℄ or if CofJNR[P ()℄g 
CofJNR[Q()℄g, then by our dis
ussion in se
tion 2, NR[P ()℄  NR[Q()℄. Let us
further assume NR[P ()℄ and NR[Q()℄ are bounded. Then for any (
0 ;
1 : : : ;
m )
in JNR[P ()℄, the equation
m m + : : : +
1  +
0 = 0 has at least one root in every
onne
ted
omponent of NR[P ()℄. A similar statement is true for JNR[Q()℄
and the
onne
ted
omponents of NR[Q()℄. Thus NR[Q()℄ has at most as many
onne
ted
omponents as NR[P ()℄.
In view of the above,
onsider now two diagonal matrix polynomials D1 () and
D2 () su
h that

(5)

JNR[D1 ()℄  JNR[P ()℄  JNR[D2 ()℄:

Let #NR[P ()℄ denote the number of
onne
ted
omponents of NR[P ()℄. From the
pre
eding dis
ussion it follows that
(6)

#NR[D1 ()℄  #NR[P ()℄  #NR[D2 ()℄:

Observe that the orders of D1 () and D2 () are irrelevant as long as (5) is satised.
Moreover, (6) gives an approximation of the number of
onne
ted
omponents of
NR[P ()℄ that
an be quite useful, espe
ially when the order n of P () is a lot
greater than its degree m.
We do know that the joint numeri
al range of a diagonal matrix polynomial is a
onvex polyhedron; see [1℄ and [9℄. Conversely, if K  C m+1 is a
onvex polyhedron
with n verti
es
(
k;0 ;
k;1 ; : : : ;
k;m ); k = 1; 2; : : : ; n;

ELA
28

P. J. Psarrakos and M. J. Tsatsomeros

then K = JNR[D()℄, where

D() = diag(
1;m ; : : : ;
n;m )m + : : : + diag(
1;1 ; : : : ;
n;1 ) + diag(
1;0 ; : : : ;
n;0 ):
Lemma 4.1. Let

T

Cm

be a nonempty
ompa
t set and

W

a subspa
e of

Cm

W \ K 6= ; for every
onvex polyhedron K 
that
ontains T . Then
CofT g \ W 6= ;.
Proof. Let K be the family of all
onvex polyhedra that
ontain T . Noti
e that
K is interse
tion
losed. Sin
e T is
ompa
t, we have that
Cm

su
h that

CofT g =

\

K 2K

K:

Dene F = fW \ K : K 2 Kg. By assumption, all members of F are
ompa
t,
nonempty and
onvex. Moreover, if W \ Kj 2 F (j = 1; 2; : : : ; p), we have

\p

j =1

T

(W

\ Kj ) = W \ (

\p

j =1

Kj ) 2 F

sin
eT pj=1 Kj 2 K. Hen
e by Helly's Theorem (see Danzer,
T Grunbaum, and Klee
[3℄), K 2K (W \ K ) 6= ;. It follows that CofT g \ W = ( K 2K K ) \ W 6= ;:
The following result is a generalization of [9, Theorem 1.1℄.
m
Theorem 4.2. Let P () = Am  + : : : + A1  + A0 be an n  n matrix polynomial.
Then

[

(7)

NR[D1 ()℄ = NR[P ()℄ =

\

NR[D2 ()℄;

where the union [interse
tion℄ is taken over all diagonal matrix polynomials

m for whi
h JNR[D1 ()℄  JNR[P ()℄  JNR[D2 ()℄.
Proof. For every diagonal matrix polynomial D1 () with
JNR[D1 ()℄  JNR[P ()℄ it is
lear that NR[D1 ()℄  NR[P ()℄. Thus
D2 ()℄

[

D1 ()

of degree

[

JNR[D1 ()℄JNR[P ()℄

NR

[D1 ()℄  NR[P ()℄:

In addition, for every
= (
0 ;
1 ; : : : ;
m ) 2 JNR[P ()℄ we
an take D1 () =
m Im +
: : : +
1 I +
0 I so that JNR[D1 ()℄ = f
g. Hen
e

[

JNR[D1 ()℄JNR[P ()℄

NR

[D1 ()℄  NR[P ()℄

and the rst equation in (7) holds. For the se
ond equation, noti
e that for every
diagonal polynomial matrix D2 () of degree m for whi
h JNR[D2 ()℄  JNR[P ()℄,
we have NR[D2 ()℄  NR[P ()℄. Thus
NR

[P ()℄



\

JNR[D2 ()℄JNR[P ()℄

NR

[D2 ()℄:

ELA
Joint Numeri
al Range

\

Let now

0 2

29

NR[

JNR[D2 ()℄JNR[P ()℄

D2 ()℄:

T1 (0 ) \ JNR[D2 ()℄ 6= ; for
D2 () satisfying JNR[D2 ()℄  JNR[P ()℄. Hen
e

From the dis
ussion in se
tion 2 (see (4)) it follows that
every diagonal matrix polynomial
from Lemma 4.1,

T1 (0 ) \ CofJNR[P ()℄g =
6 ;;
and from Proposition 2.1,

T1 (0 ) \ JNR[P ()℄ 6= ;.

So there exists unit

x0 2 C n su
h

that



x0 Am x0 )m
0 + : : : + (x0 A1 x0 )0 + x0 A0 x0 = 0;

(

\

whi
h in turn implies

NR[



P ()℄

NR[

JNR[D2 ()℄JNR[P ()℄

ompleting the proof.

D2 ()℄;

P ()℄ by its
onvex hull.

Observation 4.3. In (7) we
an repla
e JNR[

P ()℄

Observation 4.4. Theorem 4.2 gives a method of approximation of NR[
by numeri
al ranges of diagonal matrix polynomials.
2.1, we
an develop a se
ond method as follows.
polyhedra

fKt i g
( )

With the help of Proposition

Consider two sequen
es of
onvex

i = 1; 2 and t = m + 2; m + 3; : : :) so that Kt(i)

(

has

t verti
es,

Kt(1)  CofJNR[P ()℄g  Kt(2) (t = m + 2; m + 3; : : :);
and lastly so that both
for

i

;

Kt(i)

f

t  t diagonal
t  m + 2,

= 1 2 be

then for all

P ()℄g as t ! 1. If we let Dt(i) ()
(i)
polynomials asso
iated with Kt , respe
tively,

onverge to Co JNR[
matrix

Dt(1) ℄  NR[P ()℄  NR[Dt(2) ℄

NR[
and both NR[

Dt(i) ℄

onverge to NR[

P ()℄

as

t ! 1.

P ()℄

In other words, NR[

an

be approximated by numeri
al ranges of diagonal matrix polynomials of the same
degree and appropriately large order.
The results in Binding, Fareni
k, and Li [2℄ suggest another method of asso
iating numeri
al ranges of matrix polynomials to numeri
al ranges of diagonal matrix
polynomials via dilations.

A
knowledgments. We a
knowledge with thanks an anonymous referee for generously providing us with the
onstru
tion of the set

M

in se
tion 3. We also thank

Doug Fareni
k and Chi-Kwong Li for useful dis
ussions, and the former for bringing
to our attention Helly's theorem and referen
e [3℄.

ELA
30

P. J. Psarrakos and M. J. Tsatsomeros
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