Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 129132, 15 pages
doi:10.1155/2012/129132
Research Article
On Numerical Radius of a Matrix and Estimation of
Bounds for Zeros of a Polynomial
Kallol Paul and Santanu Bag
Department of Mathematics, Jadavpur University, Kolkata 700032, India
Correspondence should be addressed to Kallol Paul, kalloldada@yahoo.co.in
Received 31 March 2012; Accepted 5 June 2012
Academic Editor: Teodor Bulboaca
Copyright q 2012 K. Paul and S. Bag. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We obtain inequalities involving numerical radius of a matrix A ∈ Mn C. Using this result, we
find upper bounds for zeros of a given polynomial. We also give a method to estimate the spectral
radius of a given matrix A ∈ Mn C up to the desired degree of accuracy.
1. Introduction
Suppose A ∈ Mn C. Let WA, σA denote respectively the numerical range, spectrum
of A and wA, rσ A denote respectively the numerical radius, spectral radius of A, that is,
WA {Ax, x : x 1},
wA sup{|λ| : λ ∈ WA},
σA λ : λ is an eigenvalue of A ,
1.1
rσ A sup{|λ| : λ ∈ σA}.
It is well known that
i A/2 ≤ wA ≤ A.
Kittaneh 1 improved on the second inequality to prove that.
ii wA ≤ 1/2A 1/2A2 1/2
.
2 1/2
Clearly, 1/2A 1/2A second inequality of i.
≤ A so that inequality ii is sharper than the
2
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Let pz zn an−1 zn−1 · · · a1 z a0 be a monic polynomial where a0 , a1 , . . . , an−1 are
complex numbers and let
⎛
−an−1 −an−2
⎜ 1
0
⎜
⎜
1
⎜ 0
C p ⎜
⎜ 0
0
⎜
⎝ ·
·
·
·
·
·
·
·
·
·
⎞
· −a1 −a0
· 0
0 ⎟
⎟
· 0
0 ⎟
⎟
⎟
· 0
0 ⎟
⎟
· ·
· ⎠
· 1
0
1.2
be the Frobenius companion matrix of the polynomial pz. Then, it is well known that zeros
of p are exactly the eigenvalues of Cp. Considering Cp as an element of Mn C, we see
that if z is root of the polynomial equation pz 0, then
|z| ≤ rσ C p .
|z| ≤ w C p ,
1.3
Based on inequality ii, Kittaneh 1 obtained an estimation for wCp which gives an
upper bound for zeros of the polynomial pz.
In Section 1 we find numerical radius of some special class of matrices and use the
results obtained to give a better estimation of bounds for zeros of a polynomial.
2. On Numerical Radius of a Matrix
We first obtain bounds for numerical radius of a matrix in Mn C and use it to obtain
numerical radius for some special class of matrices.
Theorem 2.1. Suppose T ∈ Mn C and
T
A B
,
C D
2.1
where A ∈ Mr C, B ∈ Mr,n−r C, C ∈ Mn−r,r C and D ∈ Mn−r C. Then,
i wT ≤ 1/2wA wD wA − wD2 B C2 and
≤
1/2A2
B2
C2
ii T 2 1/2 A2 C2 − B2 − D2 2 4AB CD2 .
D2 Proof. i Let Z ∈ Cn and
Z
where X ∈ Cr and Y ∈ Cn−r with Z 1.
X
,
Y
2.2
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3
Then,
T Z, Z AX BY
X
,
AX, X BY, X CX, Y DY, Y CX DY
Y
2.3
and so
|T Z, Z| ≤ |AX, X| |DY, Y | BXY CXY .
2.4
Therefore, we have
wT ≤
sup
wAX2 wDY 2 B CXY ,
X2 Y 2 1
sup wAcos2 θ wDsin2 θ B C cos θ sin θ ,
θ∈0,2π
≤
2.5
1
wA wD wA − wD2 B C2 .
2
This completes the first part of the proof.
ii Proceeding as in i we can prove the second part. This completes the proof of the
theorem.
Remark 2.2. As an application of i in Theorem 2.1, T has another estimation by T 2 T ∗ T wT ∗ T as follows:
T 2 ≤
1
wA∗ A C∗ C wB∗ B D∗ D
2
wA∗ A C∗ C − wB∗ B D∗ D2 4A∗ B C∗ D2 .
2.6
Furuta 2 obtained numerical radius for a bounded linear operator T of the above form with
A aIr , B bA, C cA∗ , D dIn−r , and a, b, c, d ∈ R . If we consider A aIr , D dIn−r , C 0n−r,r where a, d ∈ R, then we can exactly calculate wT and T as proved in the
next theorem.
Theorem 2.3. Suppose B ∈ Mr,n−r C and
T
Then
aIr
B
.
On−r,r dIn−r
i wT 1/2|a d| a − d2 B2 and
√
ii T 1/ 2 a2 d2 B2 a2 − d2 − B2 2 4a2 B2 .
2.7
4
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Proof. i Following the method employed in the previous theorem, we can show that
1
2
2
wT ≤
|a d| a − d B .
2
2.8
We only need to show that there exists z0 , z0 1 such that |T z0 , z0 | equals the quantity
in the RHS.
Suppose B attains its norm at y with y 1.
Let z By kyt where k is a scalar. Then, z2 B2 |k|2 . Now
T z, z B
aIr
On−r,r dIn−r
By
By
,
ky
ky
2.9
so that
2
2
kB
dk
a
1 .
T z, z ·
z2 B2 k2
2.10
Thus for all scalar k, we get
wT ≥
a kB2 dk2 B2 k2
.
2.11
Case 1 d a ≥ 0. Define a function φ : R → R by
φk a kB2 dk2
B2 k2
.
2.12
Then using elementary calculus, we can show that φk attains its maximum at k0 d − a d − a2 B2 so that for z0 1/ By2 k0 y2 By k0 yt we get
1
2
2
a d a − d B .
|T z0 , z0 | 2
2.13
1
2
2
wT |a d| a − d B .
2
2.14
Thus, we get
International Journal of Mathematics and Mathematical Sciences
Case 2 d a ≤ 0. As before we can show that there exists k0 d − a −
that for z0 1/ By2 k0 y2 By k0 yt we get
5
d − a2 B2 so
1
2
2
|a d| a − d B .
|T z0 , z0 | 2
2.15
1
2
2
wT |a d| a − d B .
2
2.16
Thus in all cases, we get
This completes the proof of i.
ii The proof is similar to the earlier one.
This completes the proof of the theorem.
Using Theorem 2.3, we can find numerical radius of an idempotent matrix A, that is, a
matrix for which A2 A and also for a matrix for which A2 I.
Corollary 2.4. Suppose A ∈ Mn C with A2 A. Then
wA A 1
.
2
2
2.17
Proof. By Schur’s theorem, A is unitarily equivalent to an upper triangular matrix. So without
loss of generality, we can assume that
A
Ir
On−r,r
Br,n−r
,
0n−r
2.18
where Ir is the identity matrix, Br,n−r is any matrix. Using the last theorem, we get
wA A 1
.
2
2
2.19
Corollary 2.5. Suppose A ∈ Mn C and A2 I. Then
1
1
wA .
A 2
A
2.20
6
International Journal of Mathematics and Mathematical Sciences
Proof. A can be expressed as
A
Ir
On−r,r
Br,n−r
,
−In−r
2.21
where Ir is the identity matrix, Br,n−r is any matrix. By Theorem 2.3, we have
A 1
1
1 B2 B 4 B2 ,
2
2
1
4 B2 .
wA 2
2.22
Therefore
1
1
A2 1 B2 B 4 B2 ,
2
2
1
2
A
1
1 1/2B2 1/2B 4 B2
2.23
1
1
1 B2 − B 4 B2 .
2
2
By adding, we get
A2 1
A2
2 B2
2
1
⇒ A 4 B2
A
1
1
⇒ wA .
A 2
A
2.24
Corollary 2.6. Suppose A ∈ Mn C with A2n I. Then
1/n
1
1
n
wA ≥
.
A 2
An Proof. It follows from the fact that An 2 I and wAn ≥ wAn .
2.25
International Journal of Mathematics and Mathematical Sciences
7
3. Bounds for Zeros of Polynomials
Let pz zn an−1 zn−1 · · · a1 z a0 be a monic polynomial where a0 , a1 , . . . , an−1 are complex
numbers and let
⎛
−an−1 −an−2
⎜ 1
0
⎜
⎜
1
⎜ 0
C p ⎜
⎜ 0
0
⎜
⎝ ·
·
·
·
·
·
·
·
·
·
⎞
· −a1 −a0
· 0
0 ⎟
⎟
· 0
0 ⎟
⎟
⎟
· 0
0 ⎟
⎟
· ·
· ⎠
· 1
0
3.1
be the Frobenius companion matrix of the polynomial pz. Then, it is well known that zeros
of p are exactly the eigenvalues of Cp. Considering Cp as a linear operator on Cn , we see
that if z is root of the polynomial equation pz 0 then
|z| ≤ w C p
as σ C p ⊂ W C p ,
3.2
where σCp is the spectrum of operator Cp. Estimation of the roots of zeros of the
polynomial pz has been done by many mathematicians over the years, some of them are
mentioned below. Let λ be a root of the polynomial equation pz 0.
i Carmichael and Mason 3 proved that
1/2
.
|λ| ≤ 1 |a0 |2 |a1 |2 · · · |an−1 |2
3.3
ii Montel 4, 5 proved that
|λ| ≤ |a0 | |a0 − a1 | · · · |an−2 − an−1 | |an−1 1|,
|λ| ≤ n − 1 |a0 | |a1 | · · · |an−1 |.
3.4
iii Cauchy 3 proved that
|λ| ≤ 1 max{|a0 |, |a1 |, . . . , |an−1 |}.
3.5
iv Fujii and Kubo 6, 7 proved that
⎤
⎡
1/2
n−2
1⎣ π
2
|an−1 |⎦.
|ai |
|λ| ≤ cos
n1 2
i0
3.6
v Alpin et al. 8 proved that
|λ| ≤ max 1 |an−1 |1 |an−2 | · · · 1 |an−k |1/k .
1≤k≤n
3.7
8
International Journal of Mathematics and Mathematical Sciences
vi Kittaneh 1 proved that
! ! !
1 !
! 2 !1/2
!
!
C p !C p !
.
|λ| ≤
2
3.8
We develop an inequality involving numerical radius with the help of which we estimate
the zeros of the polynomial p. We show with examples that our estimation is better than the
estimations mentioned above.
Theorem 3.1. If λ is a zero of the polynomial pz, then
#
⎤
$
#
⎞2
⎛
$
$
n−2
⎥
$
an−1 1 ⎢
π $
$ 2π ⎝
% |α |2 ⎠ ⎥
⎢
%cos
cos
1
|λ| ≤ ⎥,
⎢
r
⎦
n 2⎣
n
n
r0
⎡
where αr 'n−r
k0 nCk −an−1 /n
k
3.9
an−k , r 0, 1, 2, . . . , n − 2.
Proof. Putting z ξ h in the polynomial equation pz zn an zn−1 · · · a2 z a1 0, we get
ξ hn an−1 ξ hn−1 · · · a1 ξ h a0 0.
3.10
Substituting h −an−1 /n, we get
ξn αn−2 ξn−2 αn−3 ξn−3 · · · α1 ξ α0 0,
3.11
'
k
where αr n−r
k0 n CK −an−1 /n an−k , r 0, 1, 2, . . . , n − 2.
Let Cξ be the Frobenius companion matrix of the polynomial qξ ξn αn−2 ξn−2 n−3
αn−3 ξ · · · α1 ξ α0 . B
Then Cξ A
C D , where A 01 , B −αn−2 , −αn−3 , . . . , −α0 1,n−1
⎛ ⎞
1
⎜0⎟
⎜ ⎟
⎜ ⎟
C ⎜· ⎟
,
⎜ ⎟
⎝· ⎠
0 n−1,1
⎛
0
⎜1
⎜
⎜
D ⎜·
⎜
⎝·
0
0
0
·
·
0
·
·
·
·
·
·
·
·
·
1
⎞
0
0⎟
⎟
⎟
0⎟
.
⎟
0⎠
0 n−1,n−1
3.12
Using Theorem 2.1, we get
1
wA wD wA − wD2 B C2
2
#
⎤
⎡
$
#
⎞2
⎛
$
$
n−2
⎥
$
π
π $
1⎢
$
⎥
⎢
⇒ wCξ ≤ ⎢cos %cos2 ⎝1 % |αr |2 ⎠ ⎥.
⎦
2⎣
n
n
r0
wCξ ≤
3.13
International Journal of Mathematics and Mathematical Sciences
9
This shows that if ξ0 is a zero of the polynomial qξ, then
#
⎤
$
#
⎞2
⎛
$
$
n−2
⎥
$
π $
π
1⎢
$
⎥
⎢
|ξ0 | ≤ ⎢cos %cos2 ⎝1 % |αr |2 ⎠ ⎥.
⎦
2⎣
n
n
r0
⎡
3.14
Thus if λ is a zero of the polynomial pz, then
#
⎤
$
#
⎞2
⎛
$
$
n−2
⎥
$
an−1 1 ⎢
π $
$ 2π ⎝
% |α |2 ⎠ ⎥
⎢
%cos
cos
1
|λ| ≤ ⎥.
⎢
r
⎦
n 2⎣
n
n
r0
⎡
3.15
This completes the proof of the theorem.
Example 3.2. Consider the polynomial equation pz z3 − 3z2 2z 0. Then the bounds
estimated by different mathematicians are as shown in Table 1.
But our estimation shows that if λ is a zero of the polynomial then |λ| ≤ 2.280776406
which is much better than all the estimations mentioned above.
The companion matrix of the polynomial after removing the second term can be
written as
⎛
⎞
0 1 0
⎝1 0 0⎠ A2,2 B2,1 .
C1,1 D1,1
0 1 0
3.16
Then using the above theorems, it is easy to show that |λ| ≤ 2.207 which is even better
estimation.
Example 3.3. Consider the polynomial equation pz z5 − 8z4 25z3 − 38z2 28z − 8 0.
Then, the bounds estimated by different mathematicians are as shown in Table 2.
But our estimation shows that if λ is a zero of the polynomial then |λ| ≤ 2.703669110
which is much better than all the estimations mentioned above.
Theorem 3.4. Let pz zn an−1 zn−1 · · · a1 z a0 having αi i 1, 2, . . . , n as zeros and for each
m
m
m
m
m
m ∈ N, pm z zn an−1 zn−1 · · · a2 z a1 z a0 is a polynomial having α2i i 1, 2, . . . , n
as zeros. If λ is a zero of the polynomial pz, then for all m
⎛ ⎡
⎤⎞1/2m
#
$
#
⎛
⎞2
$
$
2
n $
$ m ⎥⎟
⎜ 1 ⎢ m π
m 2
% a − cos π
⎟
⎢
.
⎝1 % an−k ⎠ ⎥
|λ| ≤ ⎜
⎝ 2 ⎣an−1 cos n n−1
⎦⎠
n
k2
3.17
10
International Journal of Mathematics and Mathematical Sciences
Table 1
Carmichael and Mason
Montel
Cauchy
Fujii and Kubo
Yuri, Chien and Yeh
Kittaneh
3.741657387
7
4
4.0098824
3.464101615
3.44572894
Table 2
Carmichael and Mason
Montel
Cauchy
Fujii and Kubo
Yuri, Chien and Yeh
54.60769176
215
39
32.16529279
9
Proof. We first prove the lemma which shows that the coefficients of pm z can be expressed
in terms of coefficients of pz.
Lemma 3.5. Suppose pz zn an−1 zn−1 · · · a1 za0 is a monic polynomial, where a0 , a1 , . . . , an−1
are complex numbers and αi , i 1, 2, . . . , n are the zeros of this polynomial. If p1 z zn 1
1
1
an−1 zn−1 · · · a1 za0 is the polynomial having α2i i 1, 2, . . . , n as zeros, then for r 1, 2, . . . , n:
1
ar
−1
2n−r
a2r
n−r
k
2 −1 ark ar−k ,
where an 1, ank an−k 0.
3.18
k1
Proof. We have
2 det zI − C p det zI C p
det z2 I − C p
⇒ p1 z2 pzp−z
1
1
1
⇒ z2n an−1 z2n−1 · · · a1 z2 a0
−1n zn an−1 zn−1 · · · a1 z a0
3.19
× zn − an−1 zn−1 · · · −1n−1 a1 z −1n a0 .
Comparing the coefficient of z2r , we get for r 1, 2, . . . n:
1
ar
−1
2n−r
a2r
n−r
k
2 −1 ark ar−k ,
k1
This completes the proof of lemma.
where an 1, ank an−k 0.
3.20
International Journal of Mathematics and Mathematical Sciences
11
The companion matrix of the monic polynomial pm z is
⎛
m
m
−a
−an−2
⎜ 1n−1
0
⎜
⎜
1
⎜ 0
C pm ⎜
⎜ 0
0
⎜
⎝ ·
·
0
0
·
·
·
·
·
·
m
· −a1
·
0
·
0
·
0
·
·
·
1
m ⎞
−a0
0
0
0
·
0
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎠
3.21
We have
2m
w C pm ≥ rσ C pm rσ C p
.
3.22
1/2m
rσ C p ≤ w C pm
.
3.23
So
Using Theorem 2.1, we get
⎤
⎡
#
$
#
⎞2
⎛
$ n $
2
$ m 2
$ m ⎥
1 ⎢ m π
π
⎝1 % an−k ⎠ ⎥
% an−1 − cos
w C pm ≤ ⎢
an−1 cos
⎦.
⎣
2
n
n
k2
3.24
Thus if λ is a zero of the polynomial pz, then
⎛ ⎡
⎤⎞1/2m
#
$
#
⎛
⎞2
$
$
$
2
n 2
$
⎜ 1 ⎢ m ⎥⎟
⎢a cos π % am − cos π
⎝1 % am ⎠ ⎥⎟
.
|λ| ≤ ⎜
⎝ 2 ⎣ n−1
n−1
n−k
⎦⎠
n
n
k2
3.25
This completes the proof of the theorem.
We next prove the theorem.
Theorem 3.6. Suppose pz zn an−1 zn−1 · · · a1 za0 is a monic polynomial and αi are the roots of
this equation i 1, 2, . . . , n, where a0 , a1 , . . . , an−1 are complex numbers with |α1 | > 1 > |α2 | > · · · >
m
m
m
m
|αn |. If the equation having roots α2i for i 1, 2, . . . , n is pm z zn an−1 zn−1 · · · a1 z a0 ,
then there exists m0 ∈ N such that
m
m
1 |an−k | ≤ |an−1 | whenever m ≥ m0 and for k 2, 3, . . . , n;
m
2 wCm p1/2 converges to rσ Cp.
12
International Journal of Mathematics and Mathematical Sciences
Proof. 1 We prove this for k 2 and the rest are similar.
First observe that
n
n
m
m
2m αk ≥ |α1 |2 − |αk |2 ,
k1 k2
n
n m
n
m 2m 2m 2m
2m
αj 2 |αk |2m .
αj αk ≤ |α1 |
|αk |
an−2 j / k1
k2
j/
k2
m an−1 3.26
Now in order to have
m m an−2 ≤ an−1 .
3.27
We get
|α1 |
2m
−
n
|αk |
2m
≥ |α1 |
2m
n
k2
|αk |
2m
n m
αj 2 |αk |2m ,
j/
k2
k2
3.28
that is,
2m
|α1 |
2m
≥ 1 |α1 |
(
n
)
|αk |
2m
n m
αj 2 |αk |2m ,
3.29
j/
k2
k2
that is,
'n
m
|α1 |2
m
1 |α1 |2
≥
'n
m
2
k2 |αk |
m
1 |α1 |2
j/
k2
2m
αj |αk |2m
m
1 |α1 |2
3.30
.
Clearly, this inequality holds good as the left-hand side converges to 1, but the right-hand
side converges to 0.
2 We have
⎤
⎡
#
$
#
⎛
⎞2
$ n $
2
$ m 2
$ m ⎥
1 ⎢ m π
π
% an−1 − cos
⎝1 % an−k ⎠ ⎥
w Cm p ≤ ⎢
an−1 cos
⎦,
⎣
2
n
n
k2
3.31
that is,
⎤
⎡
2
2
√
1 m m m w Cm p ≤ ⎣an−1 1 an−1 1 n − 2an−1 ⎦,
2
3.32
International Journal of Mathematics and Mathematical Sciences
13
that is,
⎤
⎡
2
√
2
1 m m m
m
m
w Cm p ≤ ⎣an−1 an−1 an−1 an−1 n − 2an−1 ⎦.
2
3.33
So we get
m
n
n αi 2
m
m 2m
2
w Cm p ≤ K an−1 ≤ K αi ≤ K|α1 |
1 .
i1
α1
i2
3.34
Now
2m
m
rσ C p
rσ Cm p ≤ w Cm p ≤ K|α1 |2
m
n αi 2
1 .
α
i2
1
3.35
Therefore,
m 1/2m
n αi 2
* +1/2m
rσ C p ≤ w Cm p
≤ |α1 | K 1 .
α1
i2
3.36
As the terms inside the bracket on the RHS converges to 1, we get the desired result.
This completes the proof of the theorem.
Application. As an application we can exactly find the spectral radius of a given matrix.
Consider a given matrix A of order n.
Step 1. We first find the characteristic polynomial qz zn bn−1 zn−1 · · · b1 z b0 . Suppose
αi , i 1, 2, . . . , r are the distinct roots of qz 0 with |α1 | > |α2 | > · · · > |αr |.
Step 2. Find pz qz/gcdqz, q z zr ar−1 zr−1 · · · a1 z a0 . Then, roots of qz 0
m
m
m
are αi , i 1, 2, . . . , r without multiplicity. Let pm z zn an−1 zn−1 · · · a1 z a0 be the
m
polynomial having α2i for i 1, 2, . . . , r as its zeros.
Step 3. Since |α2 | < |α1 |, taking < 1/4|α1 | − |α2 |, we can see that |α2 | < |α1 − 2. Again
m
m
using a result of 9, we get |an−1 |1/2 converging to |α1 |. So for this there exists an m0 ∈ N
m
m
such that |α1 | − < |an−1 |1/2 < |α1 | for all m ≥ m0 . Therefore,
m
m 1/2
< |α1 | |α2 | < |α1 | − < an−1 ∀m ≥ m0 .
3.37
14
International Journal of Mathematics and Mathematical Sciences
Table 3
No. of iterations
0
1
2
3
4
5
m
a4
−1
−1
−1
−1.25
1.547363281
2.394094544
m
a3
0
0
−2−3
31 × 2−12
0.000119
0
m
m
a2
0
−2−4
−2−9
13 × 2−19
0
0
a1
0
0
−2−13
3 × 2−28
0
0
m
a0
−2−5
−2−10
−2−20
−2−40
0
0
rσ Cqz
2.85
2.40
2.20
2.11
2.07
2.05
m
m 1/2 0
Step 4. Let t |an−10 |
− .
'
Find sz zr rk1 ar−k /tk zr−k zr cr−1 zr−1 · · · c1 z c0 . If the roots of sz 0
are βi , then βi αi /t, i 1, 2, . . . , r and
α1 β1 > 1 > α2 β2 > β3 > · · · > βr .
t
t
3.38
Then, sz satisfies all the criterion of Theorem 3.6.
m
Step 5. The required sequence is xm twCm s1/2 , which converges to the spectral
radius of matrix A.
Example 3.7. Consider the 5th-degree polynomial qz z5 2z4 1.
By Rouche’s theorem, it is easy to see that all the roots except one are enclosed by the
simple closed curve |z| 2.
Consider sz z5 z4 1/25 and then iterating the coefficints of Qz we get the
following.
The highest absolute value of the zeros of the polynomial is 2.055 and by 5th iteration
we get 2.05. Continuing the above process, we can find the highest absolute value of the
zeros of the polynomial up to the desired degree of accuracy. The previous best result for this
is known to be 2.414 given by Alpin 8. The iterations are shown in Table 3.
Acknowledgments
The authors would like to thank the referees for their suggestions, one of the referees
suggested the inclusion of Remark 2.2 following Theorem 2.1. The research of the first author
is partially supported by PURSE-DST, Govt. of India and the research of the second author is
supported by CSIR, India.
References
1 F. Kittaneh, “A numerical radius inequality and an estimate for the numerical radius of the Frobenius
companion matrix,” Studia Mathematica, vol. 158, no. 1, pp. 11–17, 2003.
2 T. Furuta, “Applications of polar decompositions of idempotent and 2-nilpotent operators,” Linear and
Multilinear Algebra, vol. 56, no. 1-2, pp. 69–79, 2008.
3 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
4 P. Montel, “Sur quelques limites pour les modules des zéros des polynomes,” Commentarii Mathematici
Helvetici, vol. 7, no. 1, pp. 178–200, 1934.
International Journal of Mathematics and Mathematical Sciences
15
5 P. Montel, “Sur les bornes des modules des zeros des polynomes,” Tohoku Mathematical Journal, vol. 41,
1936.
6 M. Fujii and F. Kubo, “Operator norms as bounds for roots of algebraic equations,” Proceedings of the
Japan Academy, vol. 49, pp. 805–808, 1973.
7 M. Fujii and F. Kubo, “Buzano’s inequality and bounds for roots of algebraic equations,” Proceedings of
the American Mathematical Society, vol. 117, no. 2, pp. 359–361, 1993.
8 Y. A. Alpin, M.-T. Chien, and L. Yeh, “The numerical radius and bounds for zeros of a polynomial,”
Proceedings of the American Mathematical Society, vol. 131, no. 3, pp. 725–730, 2003.
9 C. A. Hutchinson, “On Graeffe’s method for the numerical solution of algebraic equations,” The
American Mathematical Monthly, vol. 42, no. 3, pp. 149–161, 1935.
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