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Matricial norms and the zeros of lacunary polynomials
LINEAR
ALGEBRA
Matricial EMERIC
ITS
APPLICATIONS
6,
(1973)
143-148
143
Norms and the Zeros of Lacunary Polynomials DEUTSCH
Polytechnic Brooklyn,
AND
Institute New
Communicated
of Brooklyn
York
by Al&on Householder
ABSTRACT A matricial norm is a generalization
of the concept of a matrix norm which assigns
to every
n x n matrix
a k x k nonnegative
number.
In this paper matricial norms are used to determine upper bounds for the
zeros of a lacunary polynomial
A matricial
with complex
norm [l] is a mapping
matrix
,u from the algebra M,
(iv)
,4A
a nonnegative
real k
x
of complex
k matrices such
axioms are satisfied:
PbA) = I444
(4 (ii) (iii)
than
coefficients.
n x n matrices into the set M,+ of nonnegative that the following
rather
+ B) d p(A)
&EC,
VAEM,;
VA, BEM,;
+ p(B)
VA, BEM,;
#MB) G P(A)@) p(A) # 0 if A # 0
(C denotes the complex field). The set M,+ is partially ordered componentwise.
If k = 1, then p is a matrix norm [4].
Denoting proved
by r(A) the spectral radius of a matrix A E M,,
~(-4) < This generalizes a well-known A particular manner.
it has been
([I], [6]) that
G4U.
property
(1)
of matrix norms
[4].
class of matricial norms can be generated in the following
For an arbitrary
$ x q matrix
B = (bij) (i = 1,. . . , $; j =
l,...,g;bij~C),denote 0
American
Elsevier Publishing
Company,
Inc., 1973
144
EMERIC
is an arbitrary
If A = ViAi=l,...,k matrix
A, where A,,,
is a matricial upper bounds
A,,, . . . , A,, are square matrices,
norm on M,
In a recent
paper
for the moduli
with complex
=
p
+
norms
have been applied
to obtain
In this note
upper bounds for the moduli of
polynomial + C,_q_lZ-4--l
c,_,p-
coefficients
then the mapping
of the zeros of polynomials.
norms to obtain
the zeros of the lacunary
of the n x n
[l].
[2], matricial
we will apply matricial
f(z)
but fixed partitioning
DEUTSCH
+ * * * + ClZ +
and with c~.+ # 0.
co
Throughout
(3)
the
paper,
m(p, N) will denote the only positive root of the equation z” -
zp-1 -
jv =
(N > 0).
0,
A table of some values of m(p, N) can be found in [3].
PROPOSITION1.
Let ko, k,, . . ., k,_,
be arbitrary
positive
numbers.
Then all the zeros of f(z) lie i?z the circle
0:
=
max{ko/kl,
k/k,, . . ., kn+lILJ,
and
P = max{lcol/ko,icll/kl,. . . Proof.
Denoting
F the companion
I
Icn-nl/L)~
D = diag(ka, k,, . . ., k,_,,
matrix
of f(z), we have
1, 1,. . ., 1) EM,,
and by
ZEROS
OF LACUNARY
D-IFD
POLYNOMIALS
145
= ...
0
-- co
0
0
_
ko
ko koo.**
0
0
0
...
0
0
0
...
0
0
0
0
. .
..
0
0
0
0
0
0
Cl
4
1
+)... 2
. . . . . . . . . . . . . . . . . . . . . . . . . . .
ooo***
0
0
0
0
**.
0
0
0
. **
L-1 ~ kn-,
.....
. .
0
0
...
o
0
... ...
0
k,_,
0
000..*
0
0
0
0
0
000.-.
0
0
1
0
0
..
....
. .
..
1
0
0
1
ooo.*-
0
0
0
...
ooo***
0
0
0
...
CZ
k, .... . cn-,-1 ___
k n-q-1 L-4
L, 0 0
___
0
........ _~ 0
____ 0 (4)
Ifp:M,
--f M,+ is the matricial norm given by Eq. (2), corresponding
to the partition shown in Eq. (4), then u
0
k,_,
0
0
-0.
0
..a
0
0
/T
0
0
0
....................... Since the eigenvalues making use of Eq. (1)
0
1
0
..*
0
0
0
o
o
1
.a.
o
o
o
0
0
0
-1.
1
0
0
0
0
0
***
0
1
0,
of D-lFD
E
M,+.
are the same as those of F, we have,
146
EMERICDEUTSCH
(5) for all zeros z of f(z).
The characteristic
dq-l
)LQ-
-
equation of @-lFD) k&3
is
= 0,
or
whence r[m(D-lFD)] equality
= CXWZ(~, k,_&‘/c(*).
(5) prove our proposition.
COROLLARY
Indeed,
together
with in-
1. All the zeros of f(z) lie in the circle
where M = max{lcal,
/3 = M.
This relation,
n
taking
jcrl, . . . , Ic~-~]}. kj = 1 (j = 0, 1,.. ., n -
q), we obtain
a = 1 and
This upper bound of the zeros of f(z) has been found by Guggen-
heimer [3]. CCJR~LLARY 2. All the zeros of f(z) lie in the circle
where y = max{Ic,JciI,
\c,/cz\,. . . , Ic~_~_~/c~_~~}(if ci = 0, take cj = 1).
Indeed, taking kj = Ic31(if cj = 0, take kj = 1) forj
= 0, 1,.. . , n. -
4,
weobtaina=yandp=l. PROPOSITION 2. Let s be an arbitrary positive number.
Then all the
zeros of f(z) lie in the circle
where 6 = max(Ic,,I/sn, Icil/sn-‘, Proof.
In Proposition
obtain cc = s and fi = SC?.
. ., Icn-~//sn>.
1, taking kj = 9-i-l
n
(j = 0, 1,.. , n -
q) we
ZEROS
OF LACUNARY
POLYNOMIALS
147
All the zeros of f(z) lie in the circle
COROLLARY 3.
Iz/ < m(q, 1) max{)cO)lin, jciJ1’(n--l), . . , (Lgll’“>. Indeed, taking s = max {l~,_~lllj}, q
we have Ic&sj onej.
for at least
Thus 6 = 1.
Since m(g, 1) < 2 for 4 > 1, this last result improves
Fujiwara’s
[5]
121< 2 max ((c,_jll’i). l
Examples. 1. Consider the equation z3 - 8.z - 3 = 0. Applying Corollary 1, we obtain jz/ < m(2, 8) = 3.38. However, Corollary 2 yields 1~1< @2(2, 512/g) = 3.03.
The zero of largest absolute value is 3.
2. Consider the equation z3 obtain
/z/ < m(2, 18) = 4.78.
32 -
18 = 0. Applying
However,
6 = 9/4 and so 1~1< 2m(2, 9/4) = 4.17.
Proposition
Corollary
1, we
2 with s = 2 gives
The zero of largest absolute value
is 3. Remark.
If the polynomial
(3) is not lacunary,
then 4 = 1 and hence
m(q, N) = m(1, N) = I + N. We reobtain thus well-known bounds for the moduli of the zeros of an arbitrary
polynomial
[5, 71.
The author acknowledges with gratitude the comments and suggestions of Professor
A. S. Householder.
REFERENCES 1 E. Deutsch,
Matricial norms, Numer. Math. 16(1970), 73-84.
2 E. Deutsch, Matricial norms and the zeros of polynomials,
Linear
Algebra
Math.
Monthly
and A$@.
6(1970), 483-489. 3 H. Guggenheimer, On a note by Q. G. Mohammad, 54-55. 4 A. S. Householder, The Theory of Matrices in New York-Toronto-London 5 M. Marden, The geometry Mathematical
Surveys,
Amer.
Amev.
Numerical
(1964). of the zeros of a polynomial Math.
Sm.
(1949).
Analysis,
in a complex
71(1964), Blaisdell, variable,
EMERIC
148 6 F. Robert,
Etude
et utilisation
de normes
vectorielles
en analyse
DEUTSCH numekique
linkaire, Thesis, UniversitC de Grenoble (1968). 7 H. S. Wilf,
Perron-Frobenius
Mat. Sot. 12(1961), 247-250. Received February,
1971
theory and the zeros of polynomials,
Proc. Amer.
ALGEBRA
Matricial EMERIC
ITS
APPLICATIONS
6,
(1973)
143-148
143
Norms and the Zeros of Lacunary Polynomials DEUTSCH
Polytechnic Brooklyn,
AND
Institute New
Communicated
of Brooklyn
York
by Al&on Householder
ABSTRACT A matricial norm is a generalization
of the concept of a matrix norm which assigns
to every
n x n matrix
a k x k nonnegative
number.
In this paper matricial norms are used to determine upper bounds for the
zeros of a lacunary polynomial
A matricial
with complex
norm [l] is a mapping
matrix
,u from the algebra M,
(iv)
,4A
a nonnegative
real k
x
of complex
k matrices such
axioms are satisfied:
PbA) = I444
(4 (ii) (iii)
than
coefficients.
n x n matrices into the set M,+ of nonnegative that the following
rather
+ B) d p(A)
&EC,
VAEM,;
VA, BEM,;
+ p(B)
VA, BEM,;
#MB) G P(A)@) p(A) # 0 if A # 0
(C denotes the complex field). The set M,+ is partially ordered componentwise.
If k = 1, then p is a matrix norm [4].
Denoting proved
by r(A) the spectral radius of a matrix A E M,,
~(-4) < This generalizes a well-known A particular manner.
it has been
([I], [6]) that
G4U.
property
(1)
of matrix norms
[4].
class of matricial norms can be generated in the following
For an arbitrary
$ x q matrix
B = (bij) (i = 1,. . . , $; j =
l,...,g;bij~C),denote 0
American
Elsevier Publishing
Company,
Inc., 1973
144
EMERIC
is an arbitrary
If A = ViAi=l,...,k matrix
A, where A,,,
is a matricial upper bounds
A,,, . . . , A,, are square matrices,
norm on M,
In a recent
paper
for the moduli
with complex
=
p
+
norms
have been applied
to obtain
In this note
upper bounds for the moduli of
polynomial + C,_q_lZ-4--l
c,_,p-
coefficients
then the mapping
of the zeros of polynomials.
norms to obtain
the zeros of the lacunary
of the n x n
[l].
[2], matricial
we will apply matricial
f(z)
but fixed partitioning
DEUTSCH
+ * * * + ClZ +
and with c~.+ # 0.
co
Throughout
(3)
the
paper,
m(p, N) will denote the only positive root of the equation z” -
zp-1 -
jv =
(N > 0).
0,
A table of some values of m(p, N) can be found in [3].
PROPOSITION1.
Let ko, k,, . . ., k,_,
be arbitrary
positive
numbers.
Then all the zeros of f(z) lie i?z the circle
0:
=
max{ko/kl,
k/k,, . . ., kn+lILJ,
and
P = max{lcol/ko,icll/kl,. . . Proof.
Denoting
F the companion
I
Icn-nl/L)~
D = diag(ka, k,, . . ., k,_,,
matrix
of f(z), we have
1, 1,. . ., 1) EM,,
and by
ZEROS
OF LACUNARY
D-IFD
POLYNOMIALS
145
= ...
0
-- co
0
0
_
ko
ko koo.**
0
0
0
...
0
0
0
...
0
0
0
0
. .
..
0
0
0
0
0
0
Cl
4
1
+)... 2
. . . . . . . . . . . . . . . . . . . . . . . . . . .
ooo***
0
0
0
0
**.
0
0
0
. **
L-1 ~ kn-,
.....
. .
0
0
...
o
0
... ...
0
k,_,
0
000..*
0
0
0
0
0
000.-.
0
0
1
0
0
..
....
. .
..
1
0
0
1
ooo.*-
0
0
0
...
ooo***
0
0
0
...
CZ
k, .... . cn-,-1 ___
k n-q-1 L-4
L, 0 0
___
0
........ _~ 0
____ 0 (4)
Ifp:M,
--f M,+ is the matricial norm given by Eq. (2), corresponding
to the partition shown in Eq. (4), then u
0
k,_,
0
0
-0.
0
..a
0
0
/T
0
0
0
....................... Since the eigenvalues making use of Eq. (1)
0
1
0
..*
0
0
0
o
o
1
.a.
o
o
o
0
0
0
-1.
1
0
0
0
0
0
***
0
1
0,
of D-lFD
E
M,+.
are the same as those of F, we have,
146
EMERICDEUTSCH
(5) for all zeros z of f(z).
The characteristic
dq-l
)LQ-
-
equation of @-lFD) k&3
is
= 0,
or
whence r[m(D-lFD)] equality
= CXWZ(~, k,_&‘/c(*).
(5) prove our proposition.
COROLLARY
Indeed,
together
with in-
1. All the zeros of f(z) lie in the circle
where M = max{lcal,
/3 = M.
This relation,
n
taking
jcrl, . . . , Ic~-~]}. kj = 1 (j = 0, 1,.. ., n -
q), we obtain
a = 1 and
This upper bound of the zeros of f(z) has been found by Guggen-
heimer [3]. CCJR~LLARY 2. All the zeros of f(z) lie in the circle
where y = max{Ic,JciI,
\c,/cz\,. . . , Ic~_~_~/c~_~~}(if ci = 0, take cj = 1).
Indeed, taking kj = Ic31(if cj = 0, take kj = 1) forj
= 0, 1,.. . , n. -
4,
weobtaina=yandp=l. PROPOSITION 2. Let s be an arbitrary positive number.
Then all the
zeros of f(z) lie in the circle
where 6 = max(Ic,,I/sn, Icil/sn-‘, Proof.
In Proposition
obtain cc = s and fi = SC?.
. ., Icn-~//sn>.
1, taking kj = 9-i-l
n
(j = 0, 1,.. , n -
q) we
ZEROS
OF LACUNARY
POLYNOMIALS
147
All the zeros of f(z) lie in the circle
COROLLARY 3.
Iz/ < m(q, 1) max{)cO)lin, jciJ1’(n--l), . . , (Lgll’“>. Indeed, taking s = max {l~,_~lllj}, q
we have Ic&sj onej.
for at least
Thus 6 = 1.
Since m(g, 1) < 2 for 4 > 1, this last result improves
Fujiwara’s
[5]
121< 2 max ((c,_jll’i). l
Examples. 1. Consider the equation z3 - 8.z - 3 = 0. Applying Corollary 1, we obtain jz/ < m(2, 8) = 3.38. However, Corollary 2 yields 1~1< @2(2, 512/g) = 3.03.
The zero of largest absolute value is 3.
2. Consider the equation z3 obtain
/z/ < m(2, 18) = 4.78.
32 -
18 = 0. Applying
However,
6 = 9/4 and so 1~1< 2m(2, 9/4) = 4.17.
Proposition
Corollary
1, we
2 with s = 2 gives
The zero of largest absolute value
is 3. Remark.
If the polynomial
(3) is not lacunary,
then 4 = 1 and hence
m(q, N) = m(1, N) = I + N. We reobtain thus well-known bounds for the moduli of the zeros of an arbitrary
polynomial
[5, 71.
The author acknowledges with gratitude the comments and suggestions of Professor
A. S. Householder.
REFERENCES 1 E. Deutsch,
Matricial norms, Numer. Math. 16(1970), 73-84.
2 E. Deutsch, Matricial norms and the zeros of polynomials,
Linear
Algebra
Math.
Monthly
and A$@.
6(1970), 483-489. 3 H. Guggenheimer, On a note by Q. G. Mohammad, 54-55. 4 A. S. Householder, The Theory of Matrices in New York-Toronto-London 5 M. Marden, The geometry Mathematical
Surveys,
Amer.
Amev.
Numerical
(1964). of the zeros of a polynomial Math.
Sm.
(1949).
Analysis,
in a complex
71(1964), Blaisdell, variable,
EMERIC
148 6 F. Robert,
Etude
et utilisation
de normes
vectorielles
en analyse
DEUTSCH numekique
linkaire, Thesis, UniversitC de Grenoble (1968). 7 H. S. Wilf,
Perron-Frobenius
Mat. Sot. 12(1961), 247-250. Received February,
1971
theory and the zeros of polynomials,
Proc. Amer.