On Numerical Radius of a Matrix and Estimation of Bounds for Zeros of a Polynomial

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Volume 2012, Article ID 129132,15pages 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,[email protected]

Received 31 March 2012; Accepted 5 June 2012

Academic Editor: Teodor Bulboaca

Copyrightq2012 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 matrixA∈MnC. 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 matrixA∈MnCup to the desired degree of accuracy.

1. Introduction

Suppose A ∈ MnC. LetWA,σAdenote respectively the numerical range, spectrum

ofAandwA,rσAdenote respectively the numerical radius, spectral radius ofA, that is,

WA {Ax, x:x1},

wA sup{|λ|:λ∈WA},

σA λ:λis an eigenvalue ofA,

rσA sup{|λ|:λ∈σA}.

1.1

It is well known that

iA/2≤wA≤ A.

Kittaneh1improved on the second inequality to prove that.

iiwA≤1/2A1/2A21/2_.

Clearly,1/2A 1/2A21/2 _{≤ A} _{so that inequality} _ii _{is sharper than the}

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Letpz zn_a

n−1zn−1· · ·a1za0be a monic polynomial wherea0, a1, . . . , an−1are

complex numbers and let

Cp

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

−an−1 −an−2 · · −a1 −a0

1 0 · · 0 0 0 1 · · 0 0 0 0 · · 0 0

· · · · · · · · 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

1.2

be the Frobenius companion matrix of the polynomialpz. Then, it is well known that zeros ofpare exactly the eigenvalues ofCp. ConsideringCpas an element ofMnC, we see

that ifzis root of the polynomial equationpz 0, then

|z| ≤wCp, |z| ≤rσ

Cp. 1.3

Based on inequalityii, Kittaneh1 obtained an estimation for wCpwhich gives an upper bound for zeros of the polynomialpz.

InSection 1we 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 MnC and use it to obtain

numerical radius for some special class of matrices.

Theorem 2.1. SupposeT ∈MnCand

T

A B C D

, 2.1

whereA∈MrC,B∈Mr,n−rC,C∈Mn−r,rCandD∈Mn−rC. Then,

iwT≤1/2wA wD

wA−wD2_B_C2_and

iiT2 _≤ ₁_/₂_A2 _B2 _C2 _D2

1/2

A2_C2_{− B}2_{− D}22₄_AB_CD2_.

Proof. iLetZ∈Cn_and

Z

X Y

, 2.2

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Then,

TZ, Z

AXBY CXDY

,

X Y

AX, XBY, XCX, YDY, Y 2.3

and so

|TZ, Z| ≤ |AX, X||DY, Y|BXYCXY. 2.4

Therefore, we have

wT≤ sup

X2_Y2₁

wAX2_wDY2_B_CXY_,

sup

θ∈0,2π

wAcos2θwDsin2θ BCcosθsinθ,

≤ 1

2

wA wD wA−wD2_B_C2

.

2.5

This completes the first part of the proof.

iiProceeding as iniwe can prove the second part. This completes the proof of the theorem.

Remark 2.2. As an application ofiinTheorem 2.1,Thas another estimation by T2

T∗_T_wT∗_T_{as follows:}

T2_≤ 1

2

wA∗_A_C∗_C_wB∗_B_D∗_D

wA∗_A_C∗_C₋_wB∗_B_D∗_D2₄_A_∗

BC∗D2

.

2.6

Furuta2obtained numerical radius for a bounded linear operatorTof 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,C0n−r,r wherea, d∈R, then we can exactly calculatewTandTas proved in the

next theorem.

Theorem 2.3. SupposeB∈Mr,n−rCand

T

aIr B

On−r,r dIn−r

. 2.7

Then

iwT 1/2|ad|

a−d2_B2_and

iiT 1/√2

a2_d2_B2

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Proof. iFollowing the method employed in the previous theorem, we can show that

wT≤ 1

2

|ad|a−d2_B2

. 2.8

We only need to show that there existsz0, z0 1 such that|Tz0, z0|equals the quantity

in the RHS.

SupposeBattains its norm atywithy1.

Letz By kytwherekis a scalar. Then,z2_B2_|k|2_{. Now}

Tz, z

aIr B

On−r,r dIn−r

By

ky

,

By

ky

2.9

so that

Tz, z ·

1

z2

akB2_dk2

B2

k2 .

2.10

Thus for all scalark, we get

wT≥

akB2

dk2

B2_k2 .

2.11

Case 1da≥0. Define a functionφ:R → Rby

φk akB2dk2 B2

k2 . 2.12

Then using elementary calculus, we can show thatφkattains its maximum atk0 d−a

d−a2_B₂_{so that for}

z0 1/

By2_k

0y2By k0ytwe get

|Tz0, z0|

1 2

ad

a−d2_B2

. 2.13

Thus, we get

wT 1

2

|ad|a−d2_B2

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Case 2da≤0. As before we can show that there existsk0 d−a−

d−a2_B2_so

that forz0 1/

By2_k

0y2By k0ytwe get

|Tz0, z0| 1

2

|ad|a−d2_B2

. 2.15

Thus in all cases, we get

wT 1

2

|ad|a−d2_B2

. 2.16

This completes the proof ofi.

iiThe proof is similar to the earlier one. This completes the proof of the theorem.

UsingTheorem 2.3, we can find numerical radius of an idempotent matrixA, that is, a matrix for whichA2Aand also for a matrix for whichA2I.

Corollary 2.4. SupposeA∈MnCwithA2A. Then

wA A

2 1

2. 2.17

Proof. By Schur’s theorem,Ais unitarily equivalent to an upper triangular matrix. So without

loss of generality, we can assume that

A

Ir Br,n−r

On−r,r 0n−r

, 2.18

whereIris the identity matrix,Br,n−r is any matrix. Using the last theorem, we get

wA A

2 1

2. 2.19

Corollary 2.5. SupposeA∈MnCandA2I. Then

wA 1

2

A _A1

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Proof. Acan be expressed as

A

Ir Br,n−r

On−r,r −In−r

, 2.21

whereIris the identity matrix,Br,n−r is any matrix. ByTheorem 2.3, we have

A

11 2B

2 1

2B

4B2,

wA 1

2

4B2.

2.22

Therefore

A2₁1

2B

21

2B

4B2,

1

A2

1

1 1/2B2 1/2B

4B2

11 2B

2₋1

2B

4B2.

2.23

By adding, we get

A2 1

A2 2B 2

⇒

A _A1

2

4B2

⇒wA 1

2

A_A1

.

2.24

Corollary 2.6. SupposeA∈MnCwithA2nI. Then

wA≥

1 2

An 1

An

1/n

. 2.25

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3. Bounds for Zeros of Polynomials

Letpz zn_a

n−1zn−1· · ·a1za0be a monic polynomial wherea0, a1, . . . , an−1are complex

numbers and let

Cp

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

−an−1 −an−2 · · −a1 −a0

1 0 · · 0 0 0 1 · · 0 0 0 0 · · 0 0

· · · · · · · · 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

3.1

be the Frobenius companion matrix of the polynomialpz. Then, it is well known that zeros ofpare exactly the eigenvalues ofCp. ConsideringCpas a linear operator onCn_{, we see}

that ifzis root of the polynomial equationpz 0 then

|z| ≤wCp asσCp⊂WCp, 3.2

where σCp is the spectrum of operator Cp. Estimation of the roots of zeros of the polynomialpzhas been done by many mathematicians over the years, some of them are mentioned below. Letλbe a root of the polynomial equationpz 0.

iCarmichael and Mason3proved that

|λ| ≤1|a0|2|a1|2· · ·|an−1|2

1/2

. 3.3

iiMontel4,5proved that

|λ| ≤ |a0||a0−a1|· · ·|an−2−an−1||an−11|,

|λ| ≤n−1 |a0||a1|· · ·|an−1|.

3.4

iiiCauchy3proved that

|λ| ≤1max{|a0|,|a1|, . . . ,|an−1|}. 3.5

ivFujii and Kubo6,7proved that

|λ| ≤cos π

n1 1 2

⎡ ⎣

_n₋₂

i0

|ai|2

1/2

|an−1|

⎤

⎦_. 3.6

vAlpin et al.8proved that

|λ| ≤max

1≤k≤n1|an−1|1|an−2|· · ·1|an−k|

1/k

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viKittaneh1proved that

|λ| ≤ 1

2

!_!

Cp!!!!!Cp2!!!1/2

. 3.8

We develop an inequality involving numerical radius with the help of which we estimate the zeros of the polynomialp. We show with examples that our estimation is better than the estimations mentioned above.

Theorem 3.1. Ifλis a zero of the polynomialpz, then

|λ| ≤an−1

n 1 2 ⎡ ⎢ ⎢ ⎢ ⎣cos π n # $ $ $ $

%cos2π

n ⎛ ⎝₁ # $ $

%n−2

r0

|αr|2

⎞ ⎠ 2 ⎤ ⎥ ⎥ ⎥

⎦, 3.9

whereαr 'kn−r0nCk−an−1/n

k

an−k, r 0,1,2, . . . , n−2.

Proof. Puttingzξhin the polynomial equationpz zn_a

nzn−1· · ·a2za10, we get

ξhn_a

n−1ξhn−1· · ·a1ξh a0 0. 3.10

Substitutingh−an−1/n, we get

ξnαn−2ξn−2αn−3ξn−3· · ·α1ξα00, 3.11

whereαr'nk−0rnCK−an−1/nkan−k, r0,1,2, . . . , n−2.

LetCξbe the Frobenius companion matrix of the polynomialqξ ξn_α

n−2ξn−2

αn−3ξn−3· · ·α1ξα0.

ThenCξ A B C D

,whereA 0₁, B −αn−2,−αn−3, . . . ,−α01,n−1

C ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 · · 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

n−1,1

, D ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 · · 0 1 0 · · 0

· · · · 0

· · · · 0 0 0 · 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

n−1,n−1

. 3.12

UsingTheorem 2.1, we get

wCξ≤ 1

2

wA wD wA−wD2_B_C2

⇒wCξ≤ 1

2 ⎡ ⎢ ⎢ ⎢ ⎣cos π n # $ $ $ $

%cos2π

n ⎛ ⎝₁ # $ $

%n−2

r0

|αr|2

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This shows that ifξ0is a zero of the polynomialqξ, then

|ξ0| ≤

1 2 ⎡ ⎢ ⎢ ⎢ ⎣cos π n # $ $ $ $

%cos2π

n ⎛ ⎝₁ # $ $

%n−2

r0

|αr|2

⎞ ⎠ 2 ⎤ ⎥ ⎥ ⎥

⎦. 3.14

Thus ifλis a zero of the polynomialpz, then

|λ| ≤ an−1

n 1 2 ⎡ ⎢ ⎢ ⎢ ⎣cos π n # $ $ $ $

%cos2π

n ⎛ ⎝₁ # $ $

%n−2

r0

|αr|2

⎞ ⎠ 2 ⎤ ⎥ ⎥ ⎥

⎦. 3.15

This completes the proof of the theorem.

Example 3.2. Consider the polynomial equationpz z3₋₃_z2₂_z _{0. Then the bounds}

estimated by diﬀerent mathematicians are as shown inTable 1.

But our estimation shows that ifλis 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}

0 1 0

⎞

⎠A2,2 B2,1

C1,1 D1,1

. 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 equationpz z5₋₈_z4₂₅_z3₋₃₈_z2₂₈_z₋₈ _0.

Then, the bounds estimated by diﬀerent mathematicians are as shown inTable 2.

But our estimation shows that ifλis a zero of the polynomial then|λ| ≤ 2.703669110 which is much better than all the estimations mentioned above.

Theorem 3.4. Letpz zn_a

n−1zn−1· · ·a1za0havingαi i1,2, . . . , nas zeros and for each

m∈N,pmz zna_nm₋₁zn−1· · ·a2mza

m

1 za

m

0 is a polynomial havingα2

m

i i1,2, . . . , n

as zeros. Ifλis a zero of the polynomialpz, then for allm

|λ| ≤

⎛ ⎜ ⎜

⎝1₂

⎡ ⎢ ⎢

⎣a_nm₋₁cos

π n # $ $ $

%_am

n−1−cos π n 2 ⎛ ⎝₁ # $ $

%n

k2

a_nm₋_k2

⎞ ⎠ 2⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠

1/2m

.

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Table 1

Carmichael and Mason 3.741657387

Montel 7

Cauchy 4

Fujii and Kubo 4.0098824

Yuri, Chien and Yeh 3.464101615

Kittaneh 3.44572894

Table 2

Carmichael and Mason 54.60769176

Montel 215

Cauchy 39

Fujii and Kubo 32.16529279

Yuri, Chien and Yeh 9

Proof. We first prove the lemma which shows that the coeﬃcients ofpmzcan be expressed

in terms of coeﬃcients ofpz.

Lemma 3.5. Supposepz zn_a

n−1zn−1· · ·a1za0is a monic polynomial, wherea0, a1, . . . , an−1

are complex numbers and αi, i 1,2, . . . , nare the zeros of this polynomial. If p1z zn

a_n1₋₁zn−1_{· · ·}_a1

1 za

1

0 is the polynomial havingα2i i1,2, . . . , nas zeros, then forr 1,2, . . . , n:

ar1 −12n−r

a2

r2 n−r

k1

−1karkar−k

, wherean1, ankan−k0. 3.18

Proof. We have

detz2_I₋_C_p2_det

zI−CpdetzICp ⇒p1

z2pzp−z

⇒z2na_n1₋₁z2n−1· · ·a₁1z2a₀1 −1nznan−1zn−1· · ·a1za0

×zn−an−1zn−1· · · −1n−1a1z −1na0

.

3.19

Comparing the coeﬃcient ofz2r_{, we get for}_r₁_,₂_{, . . . n}_:

ar1 −12n−r

a2r2 n−r

k1

−1karkar−k

, wherean1, ankan−k0. 3.20

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The companion matrix of the monic polynomialpmzis

Cpm

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

−am

n−1 −anm−2 · · −a1m −a

m

0

1 0 · · 0 0

0 1 · · 0 0

0 0 · · 0 0

· · · ·

0 0 · · 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

. 3.21

We have

wCpm

≥rσ

Cpm

rσ

Cp2m. 3.22

So

rσ

Cp≤wCpm

1/2m

. 3.23

UsingTheorem 2.1, we get

wCpm

≤ 1 2 ⎡ ⎢ ⎢

⎣a_nm₋₁cos

π n # $ $ $

%_am

n−1−cos π n 2 ⎛ ⎝₁ # $ $

%n

k2

a_nm₋_k2

⎞ ⎠ 2⎤ ⎥ ⎥ ⎦. 3.24

Thus ifλis a zero of the polynomialpz, then

|λ| ≤

⎛ ⎜ ⎜

⎝1₂

⎡ ⎢ ⎢

⎣anm−1cos

π n # $ $ $

%_am

n−1−cos π n 2 ⎛ ⎝₁ # $ $

%n

k2

a_nm₋_k2

⎞ ⎠ 2⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠

1/2m

.

3.25

This completes the proof of the theorem. We next prove the theorem.

Theorem 3.6. Supposepz zn_a

n−1zn−1· · ·a1za0is a monic polynomial andαiare the roots of

this equationi1,2, . . . , n, wherea0, a1, . . . , an−1are complex numbers with|α1|>1>|α2|>· · ·>

|αn|. If the equation having rootsα2

m

i fori1,2, . . . , nispmz zna

m

n−1zn−1· · ·a1mza

m

0 ,

then there existsm0∈Nsuch that

1|a_nm₋_k| ≤ |a_nm₋₁|whenever m≥m0 and for k2,3, . . . , n;

2 wCmp1/2

m

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Proof. 1We prove this fork2 and the rest are similar. First observe that

a_nm₋₁n

k1

α2m

k

≥ |α1|2

m −n

k2

|αk|2

m ,

a_nm₋₂

n

j /k1

α2_jmα2_km

≤ |α1|2

m

_n

k2

|αk|2

m

n

j /k2

_α_j2m

|αk|2

m .

3.26

Now in order to have

a_nm₋₂≤a_nm₋₁. 3.27

We get

|α1|2

m −n

k2

|αk|2

m ≥ |α1|2

m

_n

k2

|αk|2

m

n

j /k2

_α_j2m

|αk|2

m

, 3.28

that is,

|α1|2

m

≥1|α1|2

m(n

k2

|αk|2

m

)

n

j /k2

_α_j2m

|αk|2

m

, 3.29

that is,

|α1|2

m

1|α1|2

m ≥

'n

k2|αk|

2m

1|α1|2

m

'_n

j /k2αj2

m |αk|2

m

1|α1|2

m . 3.30

Clearly, this inequality holds good as the left-hand side converges to 1, but the right-hand side converges to 0.

2We have

wCm

p≤ 1

2

⎡ ⎢ ⎢

⎣anm−1cos

π n # $ $ $

%_am

n−1−cos π n 2 ⎛ ⎝1 # $ $

%n

k2

a_nm₋_k2

⎞ ⎠ 2⎤ ⎥ ⎥ ⎦, 3.31 that is,

wCm

p≤ 1

2

⎡

⎣a_nm₋₁1a_nm₋₁2 1√n−2a_nm₋₁

2⎤

(13)

that is,

wCm

p≤ 1

2

⎡

⎣_am

n−1anm−1

a_nm₋₁2 a_nm₋₁√n−2a_nm₋₁

2⎤

⎦_. _3.33

So we get

wCm

p≤Ka_nm₋₁≤K

n

i1

α2m

i

≤K|α1|2

m

1

n

i2

αi

α1

2

m

. _3.34

Now

rσ

Cp2m rσ

Cm

p≤wCm

p≤K|α1|2

m

1

n

i2

αi

α1

2m

. _3.35

Therefore,

rσ

Cp≤*wCm

p+1/2m ≤ |α1|

K

1

n

i2

αi

α1

2m

1/2m

. 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 matrixAof ordern.

Step 1. We first find the characteristic polynomialqz zn_b

n−1zn−1· · ·b1zb0. Suppose

αi, i1,2, . . . , rare the distinct roots ofqz 0 with|α1|>|α2|>· · ·>|αr|.

Step 2. Findpz qz/gcdqz, qz zr_a

r−1zr−1· · ·a1za0. Then, roots ofqz 0

areαi, i 1,2, . . . , r without multiplicity. Letpmz zna_nm₋₁zn−1· · ·a₁mza0m be the

polynomial havingα2m

i fori1,2, . . . , ras its zeros.

Step 3. Since|α2| < |α1|, taking < 1/4|α1| − |α2|, we can see that|α2| < |α1 −2 . Again

using a result of9, we get|a_nm₋₁|1/2m

converging to|α1|. So for this there exists anm0 ∈N

such that|α1| − <|anm−1|1/2

m

<|α1| for allm≥m0. Therefore,

|α2| <|α1| − <a_nm₋₁

1/2m

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Table 3

No. of iterations a4m a

m

3 a

m

2 a

m

1 a

m

0 rσCqz

0 −1 0 0 0 −2−5 _2.85

1 −1 0 −2−4 ₀ ₋₂−10 _2.40

2 −1 −2−3 ₋₂−9 ₋₂−13 ₋₂−20 _2.20

3 −1.25 31×2−12 ₁₃_×₂−19 ₃_×₂−28 ₋₂−40 _2.11

4 1.547363281 0.000119 0 0 0 2.07

5 2.394094544 0 0 0 0 2.05

Step 4. Lett|am0

n−1|

1/2m₀

− . Findsz zr'r

k1ar−k/tkzr−kzrcr−1zr−1· · ·c1zc0. If the roots ofsz 0

areβi, thenβiαi/t, i1,2, . . . , rand

_β₁α1

t

>1>α2

t

β2>β3>· · ·>βr. 3.38

Then,szsatisfies all the criterion ofTheorem 3.6.

Step 5. The required sequence is xm twCms1/2

m

, which converges to the spectral radius of matrixA.

Example 3.7. Consider the 5th-degree polynomialqz z5₂_z4_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.

Considersz z5_z4 ₁_/₂5_{and then iterating the coe}_ﬃ_{cints 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 Alpin8. The iterations are shown inTable 3.

Acknowledgments

The authors would like to thank the referees for their suggestions, one of the referees suggested the inclusion ofRemark 2.2followingTheorem 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

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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.

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