EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6590
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Spectral Radius Inequalities for Certain Matrices

Mona Sakkijha1, Shatha Hasan2,∗, Maryam M Alholi3

1 Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942,
Jordan
2 Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun
26816, Jordan
3 Department of Applied Taibah University, Al Ula, Saudi Arabia

Abstract. In this paper we present upper bounds for spectral radius inequalities of 2 × 2 block
accretive-dissipative matrices.

2020 Mathematics Subject Classifications: 15A45, 47B44, 47A30, 47A63
Key Words and Phrases: Spectral radius, Accretive-Dissipative matrices, Spectral norm, Uni-
tary matrix, Cartesian decomposition, Positive semidefinite

1. Introduction

The study of matrix theory has become more and more popular in the last few decades.
Researchers are attracted to this subject because of its connections with other pure and
applied areas.
In particular, the eigenvalues are crucial in solving systems of differential equations, ana-
lyzing population growth models and calculating powers of matrices. It is not always easy
to calculate the eigenvalues. However, in many scientific problems it is enough to know
that the eigenvalues lie in specific region. Such information is provided by comparing
between spectral radius and unitarily invariant norms.
A large number of inequalities involving spectral radius in addition to matrix norm were
studied in many books that is concerning with inequalities, like Bhatia, 2007 [1]. Some
investigations on norm and spectral radius inequalities were obtained by Kittaneh in 2005
[2], Elhaddad and Kittaneh in 2007[3]. In 2015, Abu-Omar and Kittaneh studied similar
topics; they applied spectral radius and norm inequalities for any two by two block ma-
trices [4].
Moreover, in 2025, Sakkijha and Hasan studied sum, difference and commutators for spec-
tral radius inequalities involving accretive-dissipative matrices [5].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6590

Email addresses: m.sakkijha@ju.edu.jo (M. Sakkijha),
shatha@bau.edu.jo (S. Hasan), Mholi@taibahu.edu.sa (M. Alholi)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 2 of 9

The spectral radius r(X) of a matrix X ∈Mn(C) is defined as

r(X) = max{|λ| : λ ∈ σ(X)}. (1)

It’s well known that
r(X) ⩽ ∥X∥ for every X ∈Mn(C), (2)

If X is positive semidefinite, then

r(X) = ∥X∥, (3)

where ∥X∥ is the spectral norm of X which is defined as max∥ν∥=1∥Xν∥ and satisfies

∥X∥ = ∥X∗∥ = ∥X∗X∥1/2 = ∥XX∗∥1/2 . (4)

Moreover, for any X,Υ ∈Mn(C), δ ∈ C and a positive integer n,

r(δX) = |δ|r(X). (5)

A special case of the spectral mapping theorem, which asserts that

r (Xn) = rn(X). (6)

In addition, two useful facts that are related to the spectral radius are as follows:

r (X∗) = r(X) (7)

and
r (UXU∗) = r(X) (8)

for every unitary matrix U, i.e. U ∗ U = I,
A commutative property which asserts that

r(XΥ) = r(ΥX). (9)

The last property is an immediate consequence of the fact that the spectra of the operators
XΥ and ΥX have the same nonzero elements.
A matrix Ψ ∈ Mn(C) is referred to as positive semidefinite (p.s.d.) matrix if (Ψν, ν) ≥
0 ∀ν ∈ Cn. It is called accretive-dissipative (Acc-Dis) if in its Cartesian decomposition
(CD) Ψ = ψ1 + iψ2, the matrices ψ1 = Re(Ψ) = Ψ+Ψ∗

2 and ψ2 = Im(Ψ) = Ψ−Ψ∗

2i are p.s.d.
Accretive-dissipative matrices found many applications. For example, Gunzburger and
Plemmons used the results in their study of energy conserving norms for the solution of
hyperbolic systems of partial differential equations, see [6]. Many researchers are interested
with this kind of Matrices like, George and Ikramor (2005)[7], also Sakkijha and Hasan
(2024)[8].



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 3 of 9

2. Basic Lemmas

In order to prove our main results, we need the following lemmas.

Lemma 1. [9] If ψ1, ψ2 ∈Mn(C) are p.s.d, then

∥ψ1 + iψ2∥ ⩽ ∥ψ1 + ψ2∥.

Lemma 2. [10] If ψ1, ψ2 ∈Mn(C) are p.s.d, then

∥ψ1 + ψ2∥ ⩽ max(∥ψ1∥, ∥ψ2∥) +
∥∥∥ψ1/2

1 ψ
1/2
2

∥∥∥ .
Lemma 3. [11] If ψ1, ψ2 ∈Mn(C) are p.s.d, then

∥ψ1ψ2 − ψ2ψ1∥ ⩽
1

2
∥ψ1∥∥ψ2∥.

Lemma 4. [12] If ψ1, ψ2, ψ3, ψ4 ∈Mn(C), then

r

([
ψ1 ψ2

ψ3 ψ4

])
⩽ r

([
∥ψ1∥ ∥ψ2∥
∥ψ3∥ ∥ψ4∥

])
.

Lemma 5. [13] If ψ1, ψ2 ∈Mn(C), then∥∥∥∥[ ψ1 0
0 ψ2

]∥∥∥∥ = max(∥ψ1∥, ∥ψ2∥).

Lemma 6. [14] If ψ1, ψ2 ∈Mn(C) are p.s.d, then∥∥∥ψ1/2
1 ψ

1/2
2

∥∥∥ ⩽ ∥ψ1ψ2∥1/2.

Lemma 7. [10] If ψ1, ψ2 ∈Mn(C) are p.s.d, then

∥ψ1 − ψ2∥ ≤ max(∥ψ1∥, ∥ψ2∥).

3. Main Results

In this section, we will present some spectral radius inequalities for 2×2 block accretive-
dissipative matrices.

Theorem 1. Let Ψ,Φ ∈ Mn(C) be Acc-Dis matrices with CD Ψ = ψ1 + iψ2 and Φ =
ϕ1 + iϕ2. Then

r

([
Ψ 0
Φ 0

])
⩽

√
max (r2(ψ1), r2(ψ2)) + 2r(ψ1)r(ψ2) = α.



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 4 of 9

Proof.

Consider r2
([

Ψ 0
Φ 0

])
= r

([
Ψ 0
Φ 0

])2

(by(6))

= r

([
Ψ2 0
ΦΨ 0

])
= r

([
Ψ 0
Φ 0

] [
Ψ 0
0 0

])
= r

([
Ψ 0
0 0

] [
Ψ 0
Φ 0

])
(by(9))

= r

([
Ψ2 0
0 0

])
⩽ r

([ ∥∥Ψ2
∥∥ 0

0 0

])
(by Lemma 4)

= r

([ ∥∥(ψ2
1 − ψ2

2

)
+ i(ψ1ψ2 + ψ2ψ1)

∥∥ 0
0 0

])
=

∥∥(ψ2
1 − ψ2

2

)
+ i(ψ1ψ2 + ψ2ψ1)

∥∥
⩽

∥∥ψ2
1 − ψ2

2

∥∥+ ∥ψ1ψ2 + ψ2ψ1∥
⩽

∥∥ψ2
1 − ψ2

2

∥∥+ ∥ψ1ψ2∥+ ∥ψ2ψ1∥
⩽ max

(∥∥ψ2
1

∥∥ , ∥∥ψ2
2

∥∥)+ ∥ψ1ψ2∥+ ∥ψ2ψ1∥ (by Lemma 7)

≤ max
(
r2(ψ1), r

2(ψ2)
)
+ 2r(ψ1)r(ψ2) (by (6) and (9)).

The proof is obvious by taking the square root.

Theorem 2. Let Ψ,Φ ∈Mn(C) be Acc-Dis matrices with CD Ψ = ψ1+ iψ2,Φ = ϕ1+ iϕ2.
Then

r

([
Ψ 0
Φ 0

])
⩽

√
max (r2(ψ1), r2(ψ2)) + max (r2(ϕ1), r2(ϕ2)) +

3

2
r(ψ1)r(ψ2) +

3

2
r(ϕ1)r(ϕ2) = β

Proof. Consider r

([
Ψ 0
Φ 0

])
⩽

∥∥∥∥[ Ψ 0
Φ 0

]∥∥∥∥ (by (3))



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 5 of 9

=

∥∥∥∥[ Ψ∗ Φ∗

0 0

] [
Ψ 0
Φ 0

]∥∥∥∥1/2 ( by (4))

=

∥∥∥∥[ Ψ∗Ψ+Φ∗Φ 0
0 0

]∥∥∥∥1/2 = ∥Ψ∗Ψ+Φ∗Φ∥1/2

=∥(ψ1 − iψ2)(ψ1 + iψ2) + (ϕ1 − iϕ2)(ϕ1 + iϕ2)∥1/2

=
∥∥(ψ2

1 + ψ2
2

)
+
(
ϕ21 + ϕ22

)
+ i(ψ1ψ2 − ψ2ψ1) + i(ϕ1ϕ2 − ϕ2ϕ1)

∥∥1/2
⩽
(∥∥ψ2

1 + ψ2
2

∥∥+
∥∥ϕ21 + ϕ22

∥∥+ ∥ψ1ψ2 − ψ2ψ1∥+ ∥ϕ1ϕ2 − ϕ2ϕ1∥
)1/2

⩽(max(∥ψ2
1∥, ∥ψ2

2∥) + ∥ψ1ψ2∥+max(∥ϕ21∥, ∥ϕ22∥) + ∥ϕ1ϕ2∥

+
1

2
∥ψ1∥∥ψ2∥+

1

2
∥ϕ1∥∥ϕ2∥)

1
2 (by Lemma 2 and Lemma 3)

⩽(max(∥ψ2
1∥, ∥ψ2

2∥) + ∥ψ1∥∥ψ2∥+max(∥ϕ21∥, ∥ϕ22∥) + ∥ϕ1∥∥ϕ2∥

+
1

2
∥ψ1∥∥ψ2∥+

1

2
∥ϕ1∥∥ϕ2∥)

1
2

=

√
max (r2(ψ1), r2(ψ2)) + max (r2(ϕ1), r2(ϕ2)) +

3

2
r(ψ1)r(ψ2) +

3

2
r(ϕ1)r(ϕ2)

By Theorem 1 and 2, we conclude that

r

([
Ψ 0
Φ 0

])
≤ min(α, β).

Corollary 1. Let Ψ,Φ ∈ Mn(C ) be Acc-Dis matrices with CD Ψ = ψ1 + iψ2, Φ =
ϕ1 + iϕ2. Then

r

([
0 Φ
0 Ψ

])
⩽ min(α, β).

Proof. The proof is observed by using (8) as follows:

r

([
0 Φ
0 Ψ

])
= r

(
U∗

[
Ψ 0
Φ 0

]
U

)
= r

([
Ψ 0
Φ 0

])
,

where U =

[
0 I
I 0

]
.

Corollary 2. Let Ψ,Φ ∈Mn(C) be Acc-Dis matrices with CD Ψ = ψ1+iψ2, Φ = ϕ1+iϕ2.
Then

r

([
Ψ 0
−Φ 0

])
≤ min(α, β).



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 6 of 9

Proof. The proof is due to using (8), since

r

([
Ψ 0
−Φ 0

])
= r

(
U∗

[
Ψ 0
Φ 0

]
U

)
= r

([
Ψ 0
Φ 0

])
,

where U =

[
I 0
0 −I

]
.

Corollary 3. Let Φ,Ψ ∈Mn(C) be Acc-Dis matrices with CD Ψ = ψ1+iψ2,Φ = ϕ1+iϕ2.
Then

r

([
Ψ Φ
0 0

])
≤ min(α, β).

Proof. Consider r

([
Ψ Φ
0 0

])
= r

([
Ψ∗ 0
Φ∗ 0

])
(by (7))

Now, using the same procedure in Theorem 1, we get

r

([
Ψ Φ
0 0

])
⩽

(∥∥∥(Ψ∗)2
∥∥∥)1/2

=
(∥∥∥(Ψ2

)∗∥∥∥)1/2
=

(∥∥Ψ2
∥∥)1/2 ⩽ α.

Also, r

([
Ψ Φ
0 0

])
⩽

∥∥∥∥[ Ψ Φ
0 0

]∥∥∥∥ =

∥∥∥∥[ Ψ Φ
0 0

] [
Ψ∗ 0
Φ∗ 0

]∥∥∥∥1/2 ⩽ β, which completes

the proof.

Theorem 3. Let Ψ,Φ ∈Mn(C) be Acc-Dis matrices with CD Ψ = ψ1+ iψ2,Φ = ϕ1+ iϕ2.
Then

r

([
Ψ Φ
Φ Ψ

])
≤ r(ψ1 + ϕ1) + r(ψ2 + ϕ2).

Proof. Consider r

(
U∗

[
Ψ Φ
Φ Ψ

]
U

)
= r

([
Ψ Φ
Φ Ψ

])
, (by (8))

where U = 1√
2

[
I I
−I I

]
.



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 7 of 9

Now, r

([
Ψ Φ
Φ Ψ

])
=

1

2
r

([
I I
−I I

] [
Ψ Φ
Φ Ψ

] [
I −I
I I

])
=

1

2
r

([
2(Ψ + Φ) 0

0 2(Ψ− Φ)

])
= r

([
Ψ+Φ 0

0 Ψ− Φ

])
( by (5))

⩽

∥∥∥∥[ Ψ+Φ 0
0 Ψ− Φ

]∥∥∥∥ ( by (3))

=

∥∥∥∥[ (ψ1 + ϕ1) + i(ψ2 + ϕ2) 0
0 (ψ1 − ϕ1) + i(ψ2 − ϕ2)

]∥∥∥∥
=

∥∥∥∥[ (ψ1 + ϕ1) 0
0 (ψ1 − ϕ1)

]
+ i

[
(ψ2 + ϕ2) 0

0 (ψ2 − ϕ2)

]∥∥∥∥
⩽

∥∥∥∥[ (ψ1 + ϕ1) 0
0 (ψ1 − ϕ1)

]∥∥∥∥+

∥∥∥∥[ (ψ2 + ϕ2) 0
0 (ψ2 − ϕ2)

]∥∥∥∥
= max (∥ψ1 + ϕ1∥, ∥ψ1 − ϕ1∥) + max (∥ψ2 + ϕ2∥, ∥ψ2 − ϕ2∥) (by Lemma 5)

= ∥ψ1 + ϕ1∥+ ∥ψ2 + ϕ2∥, ( Since ψ1, ψ2, ϕ1, ϕ2 are p.s.d)

= r(ψ1 + ϕ1) + r(ψ2 + ϕ2) (by(3)).

Theorem 4. Let Ψ ∈Mn(C) be Acc-Dis matrix with CD Ψ = ψ1 + iψ2.Then

r

([
Ψ Ψ
−Ψ −Ψ

])
≤ 2r(ψ2 + ψ2).

Proof. Consider r

([
Ψ Ψ
−Ψ −Ψ

])
= r

(
U

[
Ψ Ψ
−Ψ −Ψ

]
U∗

)
, (by (8)),

where U = 1√
2

[
I I
−I I

]



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 8 of 9

r

([
Ψ Ψ
−Ψ −Ψ

])
=

1

2
r

([
I I
−I I

] [
Ψ Ψ
−Ψ −Ψ

] [
I −I
I I

])
=

1

2
r

([
0 0

−4Ψ 0

])
= 2r

([
0 0
Ψ 0

])
⩽ 2

∥∥∥∥[ 0 0
Ψ 0

]∥∥∥∥
= 2∥Ψ∥
= 2∥ψ1 + iψ2∥
⩽ 2∥ψ1 + ψ2∥ (by Lemma 1)

= 2r(ψ1 + ψ2) (by (3))

The following is an example of Theorem 4.

Example 1. Let Ψ =

[
1 1
1 1

]
+ i

[
1 0
0 1

]
=

[
1 + i 1
1 1 + i

]
.

Note that r

([
Ψ Ψ
−Ψ −Ψ

])
= r

(
U

[
Ψ Ψ
−Ψ −Ψ

]
U∗

)
=

1

2
r

([
0 0
−4Ψ 0

])
= 2∥Ψ∥ =

2r(Ψ), where U = 1√
2

[
I I
−I I

]
.

To find r(Ψ), we compute the eigenvalues of Ψ as follows:

det(λI −Ψ) = det

[
λ− 1− i −1
−1 λ− 1− i

]
= det

[
λ− (1 + i) −1

−1 λ− (1 + i)

]
= (λ− (1 + i))2 − 1 = 0.

⇒ (λ− (1 + i))2 = 1 ⇒ λ ∈ {2 + i, i}.
So, r(Ψ) = max |λ| =

√
4 + 1 =

√
5.

Now it is very easy to find r(ψ1 + ψ2) = r

([
2 1
1 2

])
by calculating their eigenvalues

which are 1, 3. Consequently, r(ψ1 + ψ2) = 3.
Hence, 2

√
5 = 2r(Ψ) ≤ 2r(ψ1 + ψ2) = 6 as it was concluded by Theorem 4.

4. Conclusion

New results for spectral radius of 2 × 2 block matrices involving accretive-dissipative
matrices were discussed and proved, also an example is given.



M. Sakkijha, S. Hasan, M. Alholi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6590 9 of 9

In future, our plan is going to make extension of our inequalities for accretive-dissipative
matrices, which are matrices whose only real part is p.s.d.

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