EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 806-818
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Bounds For The Eigenvalues Of Matrix
Polynomials

Aliaa Burqan1,∗, Hamdan Hbabesh1, Ahmad Qazza1, Mona Khandaqji2

1 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan
2 Department of Basic Science, Faculty of Arts and Science, Applied Science Private
University, Amman 11931, Jordan

Abstract. We employ several numerical radius inequalities to the square of the Frobenius com-
panion matrices of monic matrix polynomials to provide new bounds for the eigenvalues of these
polynomials.

2020 Mathematics Subject Classifications: 26A33, 41A58

Key Words and Phrases: Bounds for the zeroes of polynomials, companion matrix, spectral
radius, numerical radius

1. Introduction

Let Mn×m(C) stands for the algebra of all n ×m complex matrices where, n,m ∈ N.
For n = m, we may use the symbol Mn(C). For A ∈ Mn(C), let σ(A), r(A), w(A) and ∥A∥
denote the spectrum, the spectral radius, the numerical radius and the spectral norm of A,
respectively. Recall that σ(A) = {λ ∈ C : det(λI − A) = 0}, r(A) = max{|λ| : λ ∈ σ(A)},
w(A) = max

∥x∥=1
|⟨Ax, x⟩| and ∥A∥ = max{

√
λ : λ ∈ σ(A∗A)}, where A∗ = [āji] for A = [aij ],

aij ∈ C. The interesting inequality that combines these concepts is

|λ| ≤ r(A) ≤ w(A) ≤ ∥A∥,

for any λ ∈ σ(A). The polynomial eigenvalue problem (PEP), finding and locating the
eigenvalues of matrix polynomials, are very important topics in scientific computation
that has attracted the attention of many researchers [2, 3, 5, 8, 9]. The PEP appears
in a variety of problems in a wide range of applications. There are numerous examples
of physical phenomena where PEPs arise naturally such as structural mechanics, control
theory, fluid mechanics.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4706

Email addresses: aliaaburqan@zu.edu.jo (A. Burqan),
aqazza@zu.edu.jo (A.Qazza), m khandakji@asu.edu.jo (M. Khandaqji)

https://www.ejpam.com 806 © 2023 EJPAM All rights reserved.



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 807

Consider the monic polynomial P (z) = Izm + Amzm−1 + · · · + A2z + A1, with degree
m ≥ 2 and matrix coefficients Ai ∈ Mn (C) for i = 1, . . . ,m, where I is the identity matrix
in Mn (C).
The Frobenius companion block matrix of P (z) is the mn×mn matrix given by

F (P ) =


−Am −Am−1 · · · −A2 −A1

I 0 · · · 0 0

0 I
. . . 0 0

...
...

. . .
...

...
0 0 · · · I 0

 .

This matrix builds an important bridge between matrix analysis and the geometry of
polynomials.
It is known that λ is an eigenvalue of P (z) iff λ ∈ σ(F (P )) and so if λ is an eigenvalue of
P (z), then

|λ| ≤ r(F (P )) ≤ w(F (P )) ≤ ∥F (P )∥.

In order to obtain new upper bounds for the eigenvalues of P (z), we provide new estimates
for the numerical radius of F 2(P ). The references [3, 5, 7, 8] contain bounds for the
eigenvalues of matrix polynomials based on various matrix inequalities. In fact, Higham
and Tisseur [5] obtained new bounds for the eigenvalues of matrix polynomials using norm
and numerical radius inequalities. Le, Du, and Nguyen [3] established specific (upper and
lower) bounds for the eigenvalues of matrix polynomials using the norms of the coefficients
matrices of a matrix polynomial. Eigenvalue bounds can be created using l-ifications, or
lower order matrix polynomials with the same eigenvalues as a given matrix polynomial,
as demonstrated by Melman [8]. Jaradat and Kittaneh [7] , derived new numerical radius
inequalities to the Frobenius companion block matrix of P (z) and implemented it in
obtaining a new upper bound for eigenvalues of P (z).

2. Main Results

The square of the Frobenius companion block matrix of P (z) can be written as

F 2(P ) =



Bm Bm−1 · · · B3 B2 B1

−Am −Am−1 · · · −A3 −A2 −A1

I 0 . . . 0 0 0

0 I 0
...

...
...

...
...

. . . 0 0 0
0 0 0 I 0 0


,

where Bj = AmAj −Aj−1, j = 1, · · · ,m, with A0 = 0.
To obtain our first new estimate for the numerical radius of F 2(p), we need the following
lemmas. The first two Lemmas can be found in [7].



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 808

Lemma 1. Let A = [Aij ] ∈ Mm(C) be the block matrix with Aij ∈ Mni×nj(C) and∑m
i=1 ni = m, where 1 ≤ i, j ≤ m. Then w(A) ≤ w([αij ]), where

αij = w

([
0 Aij

Aji 0

])
.

In particular, αii = w(Aii) for each i = 1, 2, . . . ,m.

Lemma 2. Let Tn ∈ Mn(C) be the tridiagonal matrix given by

Tn =


0 1

2 0 . . . 0
1
2 0 1

2 . . . 0
0 1

2 0 . . . 0
...

...
. . .

. . . 1
2

0 0 . . . 1
2 0

 .

Then

w(Tn) = cos

(
π

n+ 1

)
.

The following Lemmas can be found in [4, 6] and [1], respectively.

Lemma 3. Let A =

[
a b
c d

]
. Then the spectral radius of A is

r(A) =
1

2
(a+ d+

√
(a− d)2 + 4bc).

Lemma 4. Let T = [Tij ] ∈ Mn(C) with Tkm ∈ Mkm(C). Then

w(T ) ≤ 1

2

n∑
k=1

w(Tkk) +

√√√√√w2(Tkk) +
n∑

m=1
k ̸=m

∥Tkm∥2

 .

Lemma 5. Let A =

[
a b
c d

]
. Then the spectral norm of A is

∥A∥ =

[
1

2

(
|a|2 + |b|2 + |c|2 + |d|2 +

√
(|a|2 + |c|2 − |b|2 − |d|2)2 + 4|ab̄+ cd̄|2

) 1
2

]
.

Now, we introduce our first estimate for the numerical radius of F 2(P ).



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 809

Theorem 1. An upper bound of the numerical radius of F 2(P ) can be stated as follows:

w(F 2(P )) ≤1

4
(w(Bm) + w(Am−1) + γ) + cos2

(
π

mn− 1

)
+

1

2

+
1

2

√
1

4
(w (Bm) + w (Am−1) + γ)2 +

1

2

(
α+ η +

√
(α− η)2 + 4β2

)

+
1

2

√(
2 cos2

(
π

mn− 1

)
+ 1

)2

+
1

2

(
α+ η +

√
(α− η)2 + 4β2

)
,

where

α = w2
(
T
(
I,Bm−2

))
+

1

4

m−3∑
i=1

∥Bi∥2,

η = w2(T (I, Am−3)) +
1

4
∥Am−2∥2 +

1

4

m−4∑
i=1

∥Ai∥2,

β = w (T (Bm−2, I))
∥Am−2∥

2
+ w (T (Am−3, I))

∥Bm−3∥
2

+
1

4

m−4∑
i=1

(∥Ai∥∥Bi∥) ,

γ =

√
(w (Bm)− w (Am−1))

2 + 4w2 (T (Bm−1,−Am)).

Proof. For any two matrices, C,D ∈ Mn(C), let T (C,D) =

[
0 C
D 0

]
. By applying

Lemma 1 on F 2(P ), we have w(F 2(P )) ≤ w(R), where R is mn×mn matrix given by

w(Bm) w(T (Bm−1,−Am)) w(T (Bm−2, I)) w(T (Bm−3, 0)) w(T (Bm−4, 0)) . . . w(T (B3, 0)) w(T (B2, 0)) w(T (B1, 0))
w(T (−Am, Bm−1)) w(Am−1) w(T (0, Am−2)) w(T (Am−3, I)) w(T (Am−4, 0)) . . . w(T (−A3, 0)) w(T (−A2, 0)) w(T (−A1, 0))

w(T (I,Bm−2) w(T (0, Am−2)) w(0) w(T (0, 0)) w(T (0, I)) . . . w(T (0, 0)) w(T (0, 0)) w(T (0, 0))
w(T (0, Bm−3)) w(T (I, Am−3)) w(T (0, 0)) w(0) w(T (0, 0)) . . . w(T (0, 0)) w(T (0, 0)) w(T (0, 0))
w(T (0, Bm−4)) w(T (0, Am−4)) w(T (I, 0)) w(T (0, 0)) w(0) . . . w(T (0, 0)) w(T (0, 0)) w(T (0, 0))

...
...

...
...

...
. . .

...
...

...
w(T (0, B3)) w(T (0,−A3)) w(T (0, 0)) w(T (0, 0)) w(T (0, 0)) . . . w(0) w(T (0, 0)) w(T (0, I))
w(T (0, B2)) w(T (0,−A2)) w(T (0, 0)) w(T (0, 0)) w(T (0, 0)) . . . w(T (0, 0)) w(0) w(T (0, 0))
w(T (0, B1)) w(T (0,−A1)) w(T (0, 0)) w(T (0, 0)) w(T (0, 0)) . . . w(T (I, 0)) w(T (0, 0)) w(0)





w(Bm) w(T (Bm−1,−Am)) w(T (Bm−2, I)) w(T (Bm−3, 0)) w(T (Bm−4, 0)) . . . w(T (B3, 0)) w(T (B2, 0)) w(T (B1, 0))
w(T (−Am, Bm−1)) w(Am−1) w(T (0, Am−2)) w(T (Am−3, I)) w(T (Am−4, 0)) . . . w(T (−A3, 0)) w(T (−A2, 0)) w(T (−A1, 0))

w(T (I,Bm−2) w(T (0, Am−2)) w(0) w(T (0, 0)) w(T (0, I)) . . . w(T (0, 0)) w(T (0, 0)) w(T (0, 0))
w(T (0, Bm−3)) w(T (I, Am−3)) w(T (0, 0)) w(0) w(T (0, 0)) . . . w(T (0, 0)) w(T (0, 0)) w(T (0, 0))
w(T (0, Bm−4)) w(T (0, Am−4)) w(T (I, 0)) w(T (0, 0)) w(0) . . . w(T (0, 0)) w(T (0, 0)) w(T (0, 0))

...
...

...
...

...
. . .

...
...

...
w(T (0, B3)) w(T (0,−A3)) w(T (0, 0)) w(T (0, 0)) w(T (0, 0)) . . . w(0) w(T (0, 0)) w(T (0, I))
w(T (0, B2)) w(T (0,−A2)) w(T (0, 0)) w(T (0, 0)) w(T (0, 0)) . . . w(T (0, 0)) w(0) w(T (0, 0))
w(T (0, B1)) w(T (0,−A1)) w(T (0, 0)) w(T (0, 0)) w(T (0, 0)) . . . w(T (I, 0)) w(T (0, 0)) w(0)


Using the fact that w

([
0 A
0 0

])
= w

([
0 0
A 0

])
= ∥A∥

2 , for every matrix A ∈ Mn(C),

then R is equal to



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 810

w(Bm) w(T (Bm−1,−Am)) w(T (Bm−2, I))
∥Bm−3∥

2
∥Bm−4∥

2 . . . ∥B3∥
2

∥B2∥
2

∥B1∥
2

w(T (−Am, Bm−1)) w(Am−1)
∥Am−2∥

2 w(T (Am−3, I))
∥Am−4∥

2 . . . ∥A3∥
2

∥A2∥
2

∥A1∥
2

w(T (Bm−2, I)
∥Am−2∥

2 0 0 1
2 . . . 0 0 0

∥Bm−3∥
2 w(T (Am−3, I)) 0 0 0 . . . 0 0 0

∥Bm−4∥
2

∥Am−4∥
2

1
2 0 0 . . . 0 0 0

...
...

...
...

...
. . .

...
...

...
∥B3∥
2

∥A3∥
2 0 1

2 0 . . . 0 0 1
2

∥B2∥
2

∥A2∥
2 0 0 1

2 . . . 0 0 0
∥B1∥
2

∥A1∥
2 0 0 0 . . . 1

2 0 0


.

To find w(R) we need to partition the new form of R as

R =

[
R11 R12

R21 R22

]
,

where

R11 =

[
w(Bm) w (T (Bm−1,−Am))

w (T (−Am, Bm−1)) w(Am−1)

]
,

R12 =

[
w(T (Bm−2, I))

∥Bm−3∥
2

∥Bm−4∥
2 . . . ∥B3∥

2
∥B2∥
2

∥B1∥
2

∥Am−2∥
2 w(T (Am−3, I))

∥Am−4∥
2 . . . ∥A3∥

2
∥A2∥
2

∥A1∥
2

]
,

R21 =

[
w(T (Bm−2, I))

∥Bm−3∥
2

∥Bm−4∥
2 . . . ∥B3∥

2
∥B2∥
2

∥B1∥
2

∥Am−2∥
2 w(T (Am−3, I))

∥Am−4∥
2 . . . ∥A3∥

2
∥A2∥
2

∥A1∥
2

]T
,

R22 =



0 0 1
2 0 0 . . . 0

0 0 0 1
2 0 . . . 0

1
2 0 0 . . . 1

2 . . . 0

0 1
2

. . .
. . . . . .

...
...
. . .

...

0 0 1
2

. . . . . . 0 0 1
2

...
...

...
. . . . . . 0 0 0

0 0 0 . . . 1
2 0 0


.

(mn−2)×(mn−2)

To achieve our goal, we apply Lemma 4 on the matrix R. So, we need to estimate
w(R11),w(R22), ∥R12∥ and ∥R21∥. A matrix A ∈ Mn(C) is called Hermitian if A = A∗.
Since R11 is Hermitian, Lemma 3 yields that

w(R11) = r(R11)

=
1

2

(
w(Bm) + w(Am−1) +

√
(w(Bm)− w(Am−1))

2 + 4w2(T (Bm−1,−Am)

)
.

The matrix R22 can be written as

R22 = 2

(
A2 − diag

(
1

4
,
1

2
, . . . ,

1

2
,
1

4

))
,



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 811

where

A =


0 1

2 0 . . . 0
1
2 0 1

2 . . . 0
0 1

2 0 . . . 0
... 0

...
. . . 1

2
0 0 . . . 1

2 0

 .

Thus

w (R22) ≤ 2

(
w
(
A2
)
+ w(diag

(
1

4
,
1

2
, . . . ,

1

2
,
1

4

))
.

Since A is normal and by using the fact w
(
A2
)
= w2(A), we get

w (R22) ≤ 2w2 (A) + 1.

Lemma 2 yields that

w (R22) ≤ 2 cos2
(

π

mn− 1

)
+ 1.

Now, to estimate the spectral norm of R12, R21 consider

R∗
21R21 =

[
α β
β η

]
,

where

α = w2 (T (Bm−2, I)) +
1

4

m−3∑
i=1

∥Bi∥2,

β = w (T (Bm−2, I))
∥Am−2∥

2
+ w (T (Am−3, I))

∥Bm−3∥
2

+
1

4

m−4∑
i=1

(∥Ai∥∥Bi∥),

η = w2(T (Am−3, I)) +
1

4
∥Am−2∥2 +

1

4

m−4∑
i=1

∥Ai∥2.

So,

∥R21∥2 = ∥R21R
∗
21∥ = r (R21R

∗
21) =

1

2

(
α+ η +

√
(α− η)2 + 4β2

)
.

Since R21 = RT
12, then ∥R12∥ = ∥R21∥.

Now, by applying Lemma 4, we have

w(F 2(P )) ≤ w(R) ≤ 1

2

2∑
L=1

w(Rkk) +

√√√√√w2(Rkk) +
2∑

m=1
m ̸=k

∥Rkm∥2

 .

This completes the proof.



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 812

From Theorem 1, we obtain the first new bound of the eigenvalues of P (z). In fact, if
λ is an eigenvalue of P (z), then

|λ|2 ≤ r(F 2(P )) ≤ w(F 2(P )).

On the other hand, we derive a new bound for the eigenvalues of matrix polynomials using
a similar matrix to F 2(P ).
Consider the invertible matrix

B =


I I I . . . I
0 I I . . . I
0 0 I . . . I
...

... 0
. . .

...
0 0 0 0 I

 ,

where

B−1 =


I −I 0 . . . 0
0 I −I . . . 0

0 0 I
. . . 0

...
...

...
. . . −I

0 0 0 · · · I

 .

Consider the mn×mn matrix H, where H = BF 2(P )B−1;

H=



Bm −Am + I Bm−1 −Bm +Am −Am−1 Bm−2 −Bm−1 +Am−1 −Am−2 . . . B2 −B3 +A3 −A2 B1 −B2 +A2 −A1

I −Am Am −Am−1 Am−1 −Am−2 . . . A3 −A2 − I A2 −A1

I 0 0 . . . −I 0
0 I 0 . . . −I 0

0 0
. . . . . . −I 0

...
...

... . . .
...

...
0 0 0 . . . −I 0


Since H is similar to F 2(P ), if λ is an eigenvalue of P(z), we obtain

|λ|2 ≤ r(H) ≤ w(H).

In the following, we provide an estimate of the numerical radius of H in order to extract
a new upper bound for the eigenvalues of P (z).

Theorem 2. An upper bound of the numerical radius of H can be stated as follows:

w(H) ≤ 1

2

ξ + 2 cos2
(

π

mn− 3

)
+ 1 +

2 +
√
5

4
+

√
ξ2 +

(
1

2

(
α+ η +

√
(α− η)2 + 4β2

))2

+ τ2

+

√(
2 cos2

(
π

mn− 3

)
+ 1

)2

+

(
1

2

(
α+ η +

√
(α− η)2 + 4β2

))2

+ µ

+

√√√√(2 +
√
5

4

)2

+ τ2 + µ

)
,



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 813

where

ξ =
1

2

(
w(Bm −Am+I) + w(Am −Am−1) +

√
(w(Bm −Am+I)− w(Am −Am−1))

2

+4w2(T (Bm−1 −Bm +Am −Am−1, I −Am))

)
,

α = w2 (T (Bm−2 −Bm−1 +Am−1 −Am−2, I)) +
1

4

m∑
j=6

∥Bj−3 −Bj−2 +Aj−2 −Aj−3∥2,

η =
1

4
∥Am−1 −Am−2∥2 + w2(T (Am−2 −Am−3, I) +

1

4

m∑
i=7

∥Aj−3 −Aj−4∥2,

β =
1

2
w (T (Bm−2 −Bm−1 +Am−1 −Am−2, I)) ∥Am−1 −Am−2∥

+
1

2
∥Bm−3 −Bm−2 +Am−2 −Am−3∥w(T (Am−2 −Am−3, I))

+
1

4

m∑
j=7

∥Bj−4 −Bj−3 +Aj−3 −Aj−4∥∥Aj−3 −Aj−4∥,

τ =
1

2

(
∥B2 −B3 +A3 −A2∥2

4
+

∥A3 −A2 − I∥2

4
− ∥B1 −B2 +A2 −A1∥2

4
− ∥A2 −A1∥2

4

)
√√√√ (

∥B2−B3+A3−A2∥2
4 + ∥A3−A2−I∥2

4 − ∥B1−B2+A2−A1∥2
4 − ∥A2−A1∥2

4

)2
+1

4 (∥B2 −B3 +A3 −A2∥∥B1 −B2 +A2 −A1∥+ ∥A3 −A2 − I∥∥A2 −A1∥)2 .

Proof. For any two matrices, D ∈ Mn (C), let T (C,D) =

[
0 C
D 0

]
. Applying Lemma

1 for the matrix H, we have
w (H) ≤ w (S)

where the block matrix S is given by

S =

S11 S12 S13

S21 S22 S23

S31 S32 S33

 ,

S11 =

[
w(Bm −Am + I) w(T (Bm−1 −Bm +Am −Am−1, I −Am))

w(T (I −Am , Bm−1 −Bm +Am −Am−1)) w (Am −Am−1)

]
2×2

,

S12 =




w(T (I,Bm−2 −Bm−1 +Am−1 −Am−2)) w(T (0, Am−1 −Am−2))
w(T (0, Bm−3 −Bm−2 +Am−2 −Am−3)) w(T (I, Am−2 −Am−3))

... w(T (0, Am−3 −Am−4))

...
...

w(T (0, B3 −B4 +A4 −A3)) w(T (0, A4 −A3))



t
2×(mn−4)

,



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 814

S13 =

[
w(T (B2 −B3 +A3 −A2, 0)) w(T (B1 −B2 +A2 −A1, 0))

w(T (A3 −A2 − I, 0)) w(T (A2 −A1, 0))

]
2×2

,

S21 =


w(T (I,Bm−2 −Bm−1 +Am−1 −Am−2)) w(T (0, Am−1 −Am−2))
w(T (0, Bm−3 −Bm−2 +Am−2 −Am−3)) w(T (I, Am−2 −Am−3))

... w(T (0, Am−3 −Am−4))

...
...

w(T (0, B3 −B4 +A4 −A3)) w(T (0, A4 −A3))


(mn−4)×2

,

S22 =



w(0) w(T (0.0)) w(T (0.I)) w(T (0.0)) . . . w(T (0.0)) w(T (0.0))

w(T (0.0)) w(0) w(T (0.0)) w(T (0.I))
. . . w(T (0.0)) w(T (0.0))

w(T (I.0)) w(T (0.0)) w(0) w(T (0.0))
. . . w(T (0.0)) w(T (0.0))

w(T (0.0)) w(T (I.0)) w(T (0.0)) w(0)
. . . w(T (0.I)) w(T (0.0))

w(T (0.0)) w(T (0.0)) w(T (I.0)) w(T (0.0))
. . . w(T (0.0)) w(T (0.I))

...
...

...
. . .

. . . w(0) w(T (0.0))
w(T (0.0)) w(T (0.0)) · · · w(T (0.0)) w(T (I.0)) w(T (0.0)) w(0)


(mn−4)×(mn−4)

,

S23 =


w(T (−I, 0)) w(T (0, 0))
w(T (−I, 0)) w(T (0, 0))

...
...

w(T (−I, 0)) w(T (0, 0))


(mn−4)×2

,

S31 =

[
w (T (0, B2 −B3 +A3 −A2)) w(T (0, A3 −A2 − I))
w (T (0, B1 −B2 +A2 −A1)) w(T (0, A2 −A1))

]
2×2

,

S32 =

[
w(T (0,−I)) w(T (0,−I)) · · · w(T (I,−I)) w(T (0,−I))
w(T (0, 0)) w(T (0, 0)) · · · w(T (0, 0)) w(T (I, 0))

]
2×(mn−4)

,

and

S33 =

[
w (T (−I, 0)) w (T (0,−I))
w (T (−I, 0)) w (0)

]
2×2

.

We achieve our goal by applying Lemma 4 on the matrix S. So, we need to estimate w(S11),
w(S22), w(S33), ∥S12∥, ∥S13∥, ∥S21∥∥S31∥, ∥S23∥ and ∥S32∥. Since S11 is Hermitian,
applying Lemma 3 to get w(S11) = r(S11) = ξ, where

ξ =
1

2
(w(Bm −Am + I) + w(Am −Am−1))

+
1

2

√
(w (Bm −Am + I)− w (Am −Am−1))

2 + 4w2 (T (Bm−1 −Bm +Am −Am−1, I −Am)).



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 815

Using the fact that w

([
0 A
0 0

])
= w

([
0 0
A 0

])
= ∥A∥

2 for every matrix A ∈ Mn (C), the

matrix S22 can be written as

S22 =



0 0 1
2 0 . . . 0 0

0 0 0 1
2

. . . 0 0

1
2 0 0 0

. . . 0 0

0 1
2 0 0

. . . 1
2 0

0 0 1
2 0

. . . 0 1
2

...
...

...
. . .

. . . 0 0
0 0 · · · 0 1

2 0 0


and

S33 =

[
1
2

1
2

1
2 0

]
.

So,

S22 = 2

(
A2 − diag

(
1

4
,
1

2
, . . . ,

1

2
,
1

4

))
where

A =


0 1

2 0 . . . 0
1
2 0 1

2 . . . 0
0 1

2 0 . . . 0
... 0

...
. . . 1

2
0 0 . . . 1

2 0

 .

Now,

w(S22) ≤ 2

(
w(A2) + w(diag

(
1

4
,
1

2
, . . . ,

1

2
,
1

4

))
.

Given that A is normal, thus w(A2) = w2(A). Then applying Lemma 2, to get

w (S22) ≤ 2 cos2
(

π

mn− 3

)
+ 1.

Also, we have

w(S33) = r (S33) =
2 +

√
5

4
.

Since S12 can be written as

S12 =

[
w (T (Bm−2 −Bm−1 +Am−1 −Am−2, I))

∥Bm−3−Bm−2+Am−2−Am−3∥
2 . . . . . . ∥B3−B4+A4−A3∥

2
∥Am−1−Am−2∥

2 w(T (Am−2 −Am−3, I))
∥Am−3−Am−4∥

2 . . . ∥A4−A3∥
2

]
.

and S∗
12 = St

12, we have S12S
∗
12 =

[
α β
β η

]
, where

α = w2 (T (Bm−2 −Bm−1 +Am−1 −Am−2, I)) +
1

4

m∑
j=6

∥Bj−3 −Bj−2 +Aj−2 −Aj−3∥2,



A. Burqan et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 806-818 816

η =
1

4
∥Am−1 −Am−2∥2 + w2(T (Am−2 −Am−3, I)) +

1

4

m∑
j=7

∥Aj−3 −Aj−4∥2,

and

β =
1

2
w (T (Bm−2 −Bm−1 +Am−1 −Am−2, I)) ∥Am−1 −Am−2∥

+
1

2
∥Bm−3 −Bm−2 +Am−2 −Am−3∥w(T (Am−2 −Am−3, I))

+
1

4

m∑
j=7

∥Bj−4 −Bj−3 +Aj−3 −Aj−4∥∥Aj−3 −Aj−4∥.

Applying Lemma 3 , to get

∥S12∥2 = ∥S12S
∗
12∥ = r (S12S

∗
12) =

1

2

(
α+ η +

√
(a− η)2 + 4β2

)
.

Since S12 = St
21, we have

∥s21∥2 =
1

2

(
α+ η +

√
(a− η)2 + 4β2

)
.

Now, we need to find the numerical radius of S13, S31.

S13 =

[
∥B2−B3+A3−A2∥

2
∥B1−B2+A2−A1∥

2
∥A3−A2−I∥

2
∥A2−A1∥

2

]
,

S13 =

[
∥B2−B3+A3−A2∥

2
∥A3−A2−I∥

2
∥B1−B2+A2−A1∥

2
∥A2−A1∥

2

]
,

By Lemma 1, we have τ = ∥S31∥ = ∥S13∥, where

τ =
1

2

(
∥B2 −B3 +A3 −A2∥2

4
+

∥A3 −A2 − I∥2

4
− ∥B1 −B2 +A2 −A1∥2

4
− ∥A2 −A1∥2

4

)
√√√√ (

∥B2−B3+A3−A2∥2
4 + ∥A3−A2−I∥2

4 − ∥B1−B2+A2−A1∥2
4 − ∥A2−A1∥2

4

)2
+1

4 (∥B2 −B3 +A3 −A2∥∥B1 −B2 +A2 −A1∥+ ∥A3 −A2 − I∥∥A2 −A1∥)2

To find the numerical radius of S32, we have

∥S32∥2 = ∥S32S
∗
32∥ =

∥∥∥∥mn
4 − 1 1

4
1
4

1
4

∥∥∥∥ .
So,

µ = ∥S23∥2 = ∥S32S
∗
32∥ = r(S32S

∗
32) =

1

2

mn− 3

4
+

√(
m− 5

4

)2

+
1

4

 .



REFERENCES 817

Finally, applying Lemma 4 to get

w (H) ≤ w (S) ≤ 1

2

(
w (S11) + w (S22) + w (S33) +

√
w2(S11)+}|S12∥2 + ∥S13∥2

+
√

w2(S22)+}∥S21∥2 + ∥S23∥2 ++
√

w2(S33)+}∥S31∥2 + ∥S32∥2

Thus,

w(H) ≤ 1

2

ξ + 2 cos2
(

π

mn− 3

)
+ 1 +

√
ξ2 +

(
1

2

(
α+ η +

√
(α− η)2 + 4β2

))2

+ τ2

+
2 +

√
5

4
+

√(
2 cos2

(
π

mn− 3

)
+ 1

)2

+

(
1

2

(
α+ η +

√
(α− η)2 + 4β2

))2

+ µ

+

√√√√(2 +
√
5

4

)2

+ τ2 + µ.

Conclusion

We have established new effective bounds for the eigenvalues of matrix polynomials
by employing the similarity of matrices and matrix inequalities including the numerical
radius, spectral radius and matrix norms. It is worth noting that our results can be used
in many applications in geometry and matrix analysis.

Acknowledgements

The authors express their gratitude to the dear referees, who wish to remain anony-
mous, and the editor for their helpful suggestions, which improved the final version of this
paper.

Conflicts of Interest

The authors declare no conflict of interest.

References

[1] A Abu-Omar and F Kittaneh. Numerical radius inequalities for n× n operator matrices.
Linear Algebra and its Applications, 468:18–26, 2015.

[2] A Burqan A Alsawaftah and Z Al-Zhour. New efficient and accurate bounds for ze-
ros of a polynomial based on similarity of companion complex matrices. Filomat,
37(9):2961–2968, 2023.



REFERENCES 818

[3] T H B Du C T Le and T D Nguyen. On the location of eigenvalues of matrix polyno-
mials. Oper. Matrices, 13:937–954, 2019.

[4] H Guelfen and F Kittaneh. On numerical radius inequalities for operator matrices.
Numerical Functional Analysis and Optimization, 40(11):1231–1241, 2019.

[5] N J Higham and F Tisseur. Bounds for eigenvalues of matrix polynomials. Linear
Algebra Appl., 258:5–22, 2003.

[6] R A Horn. and C R Johnson. Matrix analysis. Cambridge university press, 2012.

[7] A Jaradat and F Kittaneh. Bounds for the eigenvalues of monic matrix polynomials
from numerical radius inequalities. Advances in Operator Theory, 5(3):734–743, 2020.

[8] A Melman. Polynomial eigenvalue bounds from companion matrix polynomials. Linear
Multilinear Algebra, 67:598–612, 2019.

[9] L Qiao D Xu and W Qiu. The formally second-order BDF ADI difference/compact dif-
ference scheme for the nonlocal evolution problem in three-dimensional space. Applied
Numerical Mathematics, 172:359–381, 2022.