EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5597
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Numerical Radius Inequalities Involving 2× 2 Block
Matrices

Ahmad Al-Natoor1,∗, Fadi Alrimawi2

1 Department of Mathematics, Isra University, Amman, Jordan
2 Department of Basic Sciences, Al-Ahliyya Amman University, Amman, Jordan

Abstract. In this paper, we give several upper and lower bounds for the numerical radius of 2×2
block matrices. Several special cases of our results are given.
2020 Mathematics Subject Classifications: 47A12, 47A30, 15A60
Key Words and Phrases: Numerical radius, matrix, inequality

1. Introduction

Let Mn(C) denote the space of all n × n complex matrices. The spectral norm of a
matrix A ∈ Mn(C) is defined by

∥A∥ = max
∥x∥=1

{∥Ax∥ : x ∈ Cn}.

The numerical radius of a matrix A ∈ Mn(C) is defined by

ω(A) = max
∥x∥=1

{|⟨Ax, x⟩| : x ∈ Cn}.

In [22], the author proved that the numerical radius of a matrix A ∈ Mn(C) can be
formulated as

w (A) = max
θ∈R

∥∥∥Re(eiθA)∥∥∥ ,
where Re

(
eiθA

)
denotes the real part of the matrix eiθA. Clearly, we always have

w(A) ≤ ∥A∥ (1)

for any A ∈ Mn(C).
Many generalizations and recent related results of the numerical radius w(·) were

discussed by many authors, some of these results can be found in [3], [7], [10], [12], [11],
[9], [8], [13], [14], [15], [19], and [20].

Some basic properties of the numerical radii and the spectral norms of matrices that
we need in our paper are the following: For A,B ∈ Mn(C), we have the following relations:

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5597

Email addresses: ahmad.alnatoor@iu.edu.jo (A. Al-Natoor), f.rimawi@ammanu.edu.jo (F. Alrimawi)

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



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 2 of 9

(i) w

([
A 0
0 B

])
= max{w(A), w(B))}

(ii) w(A∗) = w(A)

(iii) ∥A∗A∥ = ∥AA∗∥ = ∥A∥2

(iv)

∥∥∥∥[ A 0
0 B

]∥∥∥∥ =

∥∥∥∥[ 0 A
B 0

]∥∥∥∥ = max{∥A∥ , ∥B∥}.

Recent results concerning inequalities can be found in [1], [2], [4], [6], [5], and [21].

2. Main results

We start with the following theorem.

Theorem 1. Let A,B,C,D ∈ Mn(C). Then

w

([
A B
C D

])
≤

√
max{∥A∗A+ C∗C∥ , ∥B∗B +D∗D∥}+ ∥A∗B + C∗D∥. (2)

Proof. We have

w

([
A B
C D

])

≤
∥∥∥∥[ A B

C D

]∥∥∥∥ (by inequality (1))

=

√∥∥∥∥[ A B
C D

]∥∥∥∥2
=

√∥∥∥∥[ A∗ C∗

B∗ D∗

] [
A B
C D

]∥∥∥∥
=

√∥∥∥∥[ A∗A+ C∗C A∗B + C∗D
B∗A+D∗C B∗B +D∗D

]∥∥∥∥ (3)

=

√∥∥∥∥[ A∗A+ C∗C 0
0 B∗B +D∗D

]
+

[
0 A∗B + C∗D

B∗A+D∗C 0

]∥∥∥∥ (4)

≤

√∥∥∥∥[ A∗A+ C∗C 0
0 B∗B +D∗D

]∥∥∥∥+

∥∥∥∥[ 0 A∗B + C∗D
B∗A+D∗C 0

]∥∥∥∥
(by the triangle inequality)

=
√

max{∥A∗A+ C∗C∥ , ∥B∗B +D∗D∥}+ ∥A∗B + C∗D∥,

as required.

Based on Theorem 1 and its proof, we have several corollaries.



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 3 of 9

Corollary 1. Let A,B ∈ Mn(C). Then

w

([
A B
0 0

])
≤

√
max{∥A∥2 , ∥B∥2}+ ∥A∗B∥. (5)

Proof. The result follows by letting C = D = 0 in inequality (2).

To state our next corollary, we need the following lemma [16].

Lemma 1. Let A,B ∈ Mn(C) be normal. Then

∥A+B∥ ≤ ∥|A|+ |B|∥ ,

where |T | is the absolute value of T ∈ Mn(C) which is defined by |T | = (T ∗T )1/2 .

Corollary 2. Let A,B,C,D ∈ Mn(C). Then

w

([
A B
C D

])
≤

√
max{∥A∗A+ C∗C + |B∗A+D∗C|∥ , ∥B∗B +D∗D + |A∗B + C∗D|∥}.

Proof. By inequality (4), we have

w

([
A B
C D

])

≤

√∥∥∥∥[ A∗A+ C∗C 0
0 B∗B +D∗D

]
+

[
0 A∗B + C∗D

B∗A+D∗C 0

]∥∥∥∥
≤

√∥∥∥∥[ A∗A+ C∗C 0
0 B∗B +D∗D

]
+

∣∣∣∣[ 0 A∗B + C∗D
B∗A+D∗C 0

]∣∣∣∣∥∥∥∥
(by Lemma 1)

=

√∥∥∥∥[ A∗A+ C∗C 0
0 B∗B +D∗D

]
+

[
|B∗A+D∗C| 0

0 |A∗B + C∗D|

]∥∥∥∥
=

√∥∥∥∥[ A∗A+ C∗C + |B∗A+D∗C| 0
0 B∗B +D∗D + |A∗B + C∗D|

]∥∥∥∥
=

√
max{∥A∗A+ C∗C + |B∗A+D∗C|∥ , ∥B∗B +D∗D + |A∗B + C∗D|∥},

as required.

Letting C = D = 0 in Corollary 2, we have the following result.

Corollary 3. Let A,B ∈ Mn(C). Then

w2

([
A B
0 0

])
≤ max{∥A∗A+ |B∗A|∥ , ∥B∗B + |A∗B|∥}.



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 4 of 9

To state the next corollary, we need the following lemma [18].

Lemma 2. Let A,B,C,D ∈ Mn(C). Then∥∥∥∥[A B
C D

]∥∥∥∥ ≤
∥∥∥∥[∥A∥ ∥B∥

∥C∥ ∥D∥

]∥∥∥∥ .
Corollary 4. Let A,B,C,D ∈ Mn(C). Then

w

([
A B
C D

])
≤

√√√√ 1
2 ∥A

∗A+ C∗C∥+ 1
2 ∥B

∗B +D∗D∥
+1

2

√
(∥A∗A+ C∗C∥ − ∥B∗B +D∗D∥)2 + 4 ∥B∗A+D∗C∥2

.

Proof. By inequality (3), we have

w

([
A B
C D

])

≤

√∥∥∥∥[ A∗A+ C∗C A∗B + C∗D
B∗A+D∗C B∗B +D∗D

]∥∥∥∥
≤

√∥∥∥∥[ ∥A∗A+ C∗C∥ ∥A∗B + C∗D∥
∥B∗A+D∗C∥ ∥B∗B +D∗D∥

]∥∥∥∥
=

√
r

([
∥A∗A+ C∗C∥ ∥A∗B + C∗D∥
∥B∗A+D∗C∥ ∥B∗B +D∗D∥

])
(where r denotes the spectral radius of matrices)

=

√√√√ 1
2 ∥A

∗A+ C∗C∥+ 1
2 ∥B

∗B +D∗D∥
+1

2

√
(∥A∗A+ C∗C∥ − ∥B∗B +D∗D∥)2 + 4 ∥B∗A+D∗C∥2

,

as required.

Corollary 5. Let C,D ∈ Mn(C). Then

w2

([
I I
C D

])
≤ 1 + max

{
∥C∥2 , ∥D∥2

}
+ ∥I + C∗D∥ .

Proof. Letting A = B = I in Theorem 1, we have

w2

([
I I
C D

])
≤ max {∥I + C∗C∥ , ∥I +D∗D∥}+ ∥I + C∗D∥

≤ 1 + max {∥C∗C∥ , ∥D∗D∥}+ ∥I + C∗D∥

= 1 +max
{
∥C∥2 , ∥D∥2

}
+ ∥I + C∗D∥ .

We need the following lemma [17] to give a lower bound for inequality (5).



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 5 of 9

Lemma 3. Let A,B,C,D ∈ Mn(C). Then

w

([
A B
C D

])
≥ max

{
w(A), w(D),

w(B + C)

2
,
w(B − C)

2

}
.

Corollary 6. Let A,B ∈ Mn(C) be positive semidefinite. Then

1

2
∥A+B∥ ≤ w

([
A B
0 0

])
.

Proof. Let U = 1√
2

[
I I
−I I

]
, where I is the identity matrix, then U is unitary.

Consequently, we have

w

([
A B
0 0

])
= w

(
U∗

[
A B
0 0

]
U

)
= w

([
A−B
2

A+B
2

A−B
2

A+B
2

])
≥ max

{
w

(
A−B

2

)
, w

(
A+B

2

)
,
w (A)

2
,
w (B)

2

}
(by Lemma 3)

=
1

2
∥A+B∥ .

Corollary 7. Let A,B ∈ Mn(C) be positive semidefinite. Then

∥A∥+ ∥B∥ ≤ w2

([
A1/2 B1/2

0 0

])
+ w2

([
B1/2 A1/2

0 0

])
. (6)

Proof. By Lemma 3, we have

w2

([
A1/2 B1/2

0 0

])
≥

[
max

{
w
(
A1/2

)
,
w
(
B1/2

)
2

}]2

= max

{
∥A∥ , 1

4
∥B∥

}
. (7)

Similarly, we have

w2

([
B1/2 A1/2

0 0

])
≥ max

{
∥B∥ , 1

4
∥A∥

}
. (8)

By adding inequalities (7) and (8) and then using the fact that max(a, b) = a+b+|a−b|
2 ,

we have

w2

([
A1/2 B1/2

0 0

])
+ w2

([
B1/2 A1/2

0 0

])



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 6 of 9

≥ max

{
∥A∥ , 1

4
∥B∥

}
+max

{
∥B∥ , 1

4
∥A∥

}
=

5

8
(∥A∥+ ∥B∥) + 1

8
(|4 ∥A∥ − ∥B∥|+ |4 ∥B∥ − ∥A∥|) (9)

≥ ∥A∥+ ∥B∥ .

In inequality (5), by replacing A and B by the positive semidefinite matrices A1/2 and
B1/2 respectively, we have

w

([
A1/2 B1/2

0 0

])
≤

√
max{∥A∥ , ∥B∥}+

∥∥A1/2B1/2
∥∥. (10)

Based on inequalities (7), (8), and (10), we have the following corollary.

Corollary 8. Let A,B ∈ Mn(C) be positive semidefinite. Then

max {∥A∥ , ∥B∥} ≤ max

{
w2

([
A1/2 B1/2

0 0

])
, w2

([
B1/2 A1/2

0 0

])}
≤ max {∥A∥ , ∥B∥}+

∥∥∥A1/2B1/2
∥∥∥ .

In particular, if A1/2B1/2 = 0, then

max {∥A∥ , ∥B∥} = max

{
w2

([
A1/2 B1/2

0 0

])
, w2

([
B1/2 A1/2

0 0

])}
.

Proof. The first inequality follows from inequalities (7) and (8).
In inequality (10), by Interchanging the roles of A1/2 and B1/2, we have

w2

([
B1/2 A1/2

0 0

])
≤ max{∥A∥ , ∥B∥}+

∥∥∥A1/2B1/2
∥∥∥ . (11)

So, we can obtain the second inequality from inequalities (10) and (11).

Corollary 9. Let A,B ∈ Mn(C) be positive semidefinite. Then

max

{
∥A∥+ ∥B∥

2
,
5

8
∥A∥ , 5

8
∥B∥

}

≤ 1

2

[
w2

([
A1/2 B1/2

0 0

])
+ w2

([
B1/2 A1/2

0 0

])]
≤ max{∥A∥ , ∥B∥}+

∥∥∥A1/2B1/2
∥∥∥ .



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 7 of 9

Proof. Using inequality (9), we have

w2

([
A1/2 B1/2

0 0

])
+ w2

([
B1/2 A1/2

0 0

])

≥ 5

8
(∥A∥+ ∥B∥) + 1

8
(|4 ∥A∥ − ∥B∥|+ |4 ∥B∥ − ∥A∥|)

=
5

8
(∥A∥+ ∥B∥) + 1

8
(|4 ∥A∥ − ∥B∥|+ |∥A∥ − 4 ∥B∥|)

≥ 5

8
(∥A∥+ ∥B∥) + 5

8
|∥A∥ − ∥B∥|

=
5

4
max {∥A∥ , ∥B∥} . (12)

Using inequalities (6) and (12) we get the first inequality. Also, the second inequality can
be obtained from inequalities (10) and (11).

We end this paper with the following result.

Theorem 2. Let A,B,C,D ∈ Mn(C). Then

w

([
A B
C D

])
≥

∥∥∥∥[ Re(A) B+C∗

2
C+B∗

2 Re(D)

]∥∥∥∥ . (13)

Proof. We have

w

([
A B
C D

])
= max

θ∈R

∥∥∥∥Re

(
eiθ

[
A B
C D

])∥∥∥∥
≥

∥∥∥∥Re

([
A B
C D

])∥∥∥∥
=

1

2

∥∥∥∥[ A B
C D

]
+

[
A∗ C∗

B∗ D∗

]∥∥∥∥
=

1

2

∥∥∥∥[ A+A∗ B + C∗

C +B∗ D +D∗

]∥∥∥∥
=

1

2

∥∥∥∥[ 2Re(A) B + C∗

C +B∗ 2Re(D)

]∥∥∥∥
=

∥∥∥∥[ Re(A) B+C∗

2
C+B∗

2 Re(D)

]∥∥∥∥ ,
as required.

By inequalities (2) and (13), we have∥∥∥∥[ Re(A) B+C∗

2
C+B∗

2 Re(D)

]∥∥∥∥ ≤ w

([
A B
C D

])
≤

√
max {∥A∗A+ C∗C∥ , ∥B∗B +D∗D∥}+ ∥A∗B + C∗D∥.



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 8 of 9

3. Conclusion

In this paper, new results related to numerical radii of block matrices were given.
Several particular cases were also given. At the end of the paper, an upper bound and a

lower bound of the numerical radius of the partitioned matrix

[
A B
C D

]
were established.

Conflict of interest. The authors declare that they have no conflict of interest.
Data availability. Not applicable.
Author’s contributions. All authors contributed to each part of this work, and they

all read and approved the final manuscript.

Acknowledgements

The authors are grateful to the reviewers for their careful reading and valuable sug-
gestions.

References

[1] Al-Natoor. Norm inequalities for functions of matrices. Heliyon, 10, 2024.
[2] Al-Natoor. Norm inequalities for product of matrices. Acta Sci. Math. (Szeged), pages

https://doi.org/10.1007/s44146–024–00121–1, 2024.
[3] A. Al-Natoor, O. Hirzallah, and F. Kittaneh. Interpolating numerical radius inequal-

ities for matrices. Adv. Oper. Theory, 23, 2024.
[4] A. Al-Natoor, O. Hirzallah, and F. Kittaneh. Singular value and norm inequalities

for products and sums of matrices. Period. Math. Hung., 88:204–217, 2024.
[5] A. Al-Natoor and F. Kittaneh. Further unitarily invariant norm inequalities for pos-

itive semidefinite matrices. Positivity, 26:Paper No. 8, 11 pp., 2022.
[6] A. Al-Natoor and F. Kittaneh. Singular value and norm inequalities for positive

semidefinite matrices. Linear Multilinear Algebra, 70:4498–4507, 2022.
[7] M.W. Alomari. Numerical radius inequalities for hilbert space operators. Complex

Anal. Oper. Theory, 15:1–19, 2021.
[8] F. Alrimawi. Some inner product inequalities with applications to numerical radius

inequalities. J. Math. Computer Sci., 38:16–24, 2025.
[9] F. Alrimawi, F.A. Abushaheen, and R. Alkhateeb. Improved lower bounds for nu-

merical radius via cartesian decomposition. J. Math. Computer Sci., 33:169–175,
2024.

[10] F. Alrimawi, O. Hirzallah, and F. Kittaneh. Norm inequalities involving the weighted
numerical radii of operators. Linear Algebra Appl., 657:127–146, 2023.

[11] F. Alrimawi and H. Kawariq. On some generalized numerical radius inequalities for
hilbert space operators. J. Math. Computer Sci., 32:257–262, 2024.

[12] F. Alrimawi, H. Kawariq, and F.A. Abushaheen. Generalized-weighted numerical
radius inequalities for schatten p-norms. International Journal of Mathematics and
Computer Science, 17:1463–1473, 2022.



A. Al-Natoor, F. Alrimawi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5597 9 of 9

[13] A. Benmakhlouf, O. Hirzallah, and F. Kittaneh. On p-numerical radii of hilbert space
operators. Linear Multilinear Algebra, 69:2813–2829, 2021.

[14] P. Bhunia and K. Paul. Some improvements of numerical radius inequalities of oper-
ators and operator matrices. Linear Multilinear Algebra, 70:1995–2013, 2022.

[15] T. Bottazzi and C. Conde. Generalized numerical radius and related inequalities.
Oper. Matrices, 15:1289–1308, 2021.

[16] J.C. Bourin. A matrix subadditivity inequality for symmetric norms. Proc. Amer.
Math. Soc., 138:495–504, 2009.

[17] O. Hirzallah, F. Kittaneh, and K. Shebrawi. Numerical radius inequalities for certain
2× 2 operator matrices. Integr. Equ. Oper. Theory, 71:129–147, 2011.

[18] J.C. Hou and H.K. Du. Norm inequalities of positive operator matrices. Integr. Equ.
Oper. Theory, 22:281–294, 1995.

[19] F. Kittaneh. A numerical radius inequality and an estimate for the numerical radius
of the frobenius companion matrix. Studia Math., 158:11–17, 2003.

[20] M. Sattari, M.S. Moslehian, and T. Yamazaki. Some generalized numerical radius
inequalities for hilbert space operators. Linear Algebra Appl., 470:216–227, 2015.

[21] V. Stojiljkovic. Hermite hadamard type inequalities involving (k-p) fractional opera-
tor with (α, h - m) - p convexity. European Journal of Pure and Applied Mathematics,
16:503–522, 2023.

[22] T. Yamazaki. On upper and lower bounds of the numerical radius and an equality
condition. Studia Math., 178:83–89, 2007.