EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6813
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Properties and Applications of Generalized Numerical
Radius in Block Matrix Structures

Raja’a Al-Naimi1,2,1, Manal Al-Labadi2, Wasim Audeh2, Jamal Oudetallah2,
Mutti-Ur Rehman3, Dheyaa Alangood4

1 Department of Mathematics, Faculty of Mathematics and Data Science,
Emirates Aviation University, United Arab Emirates
2 Department of Mathematics, Faculty of Arts and Sciences, University of Petra, Amman,
Jordan
3 Center of Research and Innovation, Asia International University, 200100, Bukhara,
Uzbekistan
4 University of Tikrit, College of Basic Education, Shirqat, Iraq

Abstract. In this paper, we prove several results that generalize fundamental properties of
numerical radius and generalized numerical radius. Among our proven inequalities, we establish
theorems for matrices A ∈ Mm(Mn), where Mm(Mn) represents the set of all m × m block
complex matrices with each block belonging to Mn(C). Furthermore, we demonstrate that if
A ∈ M2(Mn) is a positive semidefinite matrix, then w(2)(A) is positive semidefinite, and if
A ∈ Mm(M2) is a positive semidefinite matrix, then w(1)(A) is positive semidefinite. Here,
w(1)(A) denotes the first partial matrix of numerical radius and w(2)(A) denotes the second
partial matrix of numerical radius, respectively. Additionally, we develop relationships with
classical matrix parameters and establish structural theorems for block matrices.
2020 Mathematics Subject Classifications: 15A42, 15A60, 47A63, 47B15, 47B47
Key Words and Phrases: Numerical radius, generalized numerical radius, block matrices,
positive semidefinite matrices, unitarily invariant norm

1. Introduction

Matrix analysis has witnessed remarkable progress in understanding the geometric
properties of operators through the numerical radius. This functional provides valuable
insights into the field of values and offers geometric interpretations that complement

1Corresponding author.
0DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6813

Email addresses: rajaa.alnaimi@uop.edu.jo, rajaa.alnaimi@eau.ac.ae (R. Al-Naimi),
manal.allabadi@uop.edu.jo (M. Al-Labadi), waudeh@uop.edu.jo (W. Audeh),
jamal.oudetallah@uop.edu.jo (J. Oudetallah), mutti.rehman@aiu.uz (M.-U. Rehman),
dheyaa.alangood@uot.edu.iq (D. Alangood)

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



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 2 of 15

traditional spectral approaches. The numerical radius, originally studied for single ma-
trices, has found extensive applications in operator theory, quantum mechanics, and
computational mathematics.

Let Mm(Mn) denote the collection of all m×m block complex matrices where each
entry belongs to Mn(C). When dealing with the special case Mm(C), we have the
standard set of m × m complex matrices. A matrix A ∈ Mm(C) is termed positive
semidefinite, written as A ≥ 0, when xTAx ≥ 0 holds for every x ∈ C.

Given A ∈ Mm(C), we use AT , Aτ , and A∗ to represent the transpose, partial trans-
pose, and conjugate transpose operations, respectively. The absolute value is expressed
as |A| = (A∗A)1/2. We denote by σ(A) = {λ1(A), . . . , λm(A)} the complete eigenvalue
set of A arranged such that |λ1(A)| ≥ · · · ≥ |λm(A)|. The spectral radius, spectral
norm, and any unitarily invariant norm of A are represented by r(A), ‖A‖, and N(A)
respectively.

For any matrix A ∈ Mm(C), the spectral radius is given by r(A) = |λ1|, while the
spectral norm is defined through ‖A‖ = max

‖x‖=1
‖Ax‖.

The numerical radius w(A) and generalized numerical radius wN (A) of a matrix A
play central roles in our analysis. Specifically, for A ∈ Mm(C), the numerical radius is
characterized by

w(A) = max
‖x‖=1

|〈Ax, x〉|,

while the generalized numerical radius takes the form

wN (A) = max
θ∈R

N(Re(eiθA)). (1)

When we set N(·) = ‖ · ‖ in equation (1), we recover the standard numerical radius:

w(A) = max
θ∈R

‖Re(eiθA)‖. (2)

A fundamental relationship exists for normal matrices A ∈ Mn(C):

w(A) = ‖A‖ = r(A). (3)

For comprehensive treatments of numerical radius properties and generalized ver-
sions, readers are directed to [1], [2], and [3].

The connection to singular values is established through ‖A‖ = s1(A), where s1(A)
represents the largest singular value of matrix A ∈ Mm(C). These singular values follow
the ordering s1(A) ≥ s2(A) ≥ · · · ≥ sm(A) and are defined via sj(A) = λj(|A|) for
j = 1, 2, . . . ,m. Comprehensive discussions of singular value theory can be found in
[4–8].

When working with matrices A = [Ai,j ] ∈ Mm(Mn) and B = [Bi,j ] ∈ Mm(Mn) of
identical dimensions, the Hadamard product, as introduced in [9], is given by A ◦ B =
[Ai,jBi,j ]. The concept of Hadamard powers, detailed in [10], is expressed as:

A◦n = [An
i,j ], (4)



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 3 of 15

with the corresponding Hadamard inverse defined by:

A◦(−1) = [A−1
i,j ]. (5)

For matrices of the form A = [aij ]
m
i,j=1 ∈ Mm(C), reference [11] provides:

A◦(1/n) = [a
1/n
i,j ]. (6)

Research findings presented in [12] establish that for A ∈ Mm(Mn):

r(A) ≤ r([‖Ai,j‖]), ‖A‖ ≤ ‖[‖Ai,j‖]‖ and w(A) ≤ r([‖Ai,j‖]). (7)

Consider a matrix A = [[ai,jl,k]
n
l,k=1]

m
i,j=1 ∈ Mm(Mn). We can construct the associated

matrix à ∈ Mn(Mm) through à = [Gl,k]
n
l,k=1 = [[ai,jl,k]

m
i,j=1]

n
l,k=1, where each Gl,k =

[ai,jl,k]
m
i,j=1 and the relationship ˜̃A = A holds.

Building upon our previous investigations into spectral properties of block matrices
through partial eigenvalues [13], we now explore complementary geometric aspects via
numerical radius analysis. While eigenvalue-based methods focus on spectral decompo-
sition and diagonalization properties, the numerical radius approach emphasizes field of
values and geometric characterizations of matrix behavior.

In the present work, we introduce novel constructions for partial matrices involving
generalized numerical radius and standard numerical radius:

w
(1)
N (A) = [wN (Gl,k)]

n
l,k=1,

w
(2)
N (A) = [wN (Ai,j)]

m
i,j=1,

w(1)(A) = [w(Gl,k)]
n
l,k=1,

and
w(2)(A) = [w(Ai,j)]

m
i,j=1.

These constructions extend classical numerical radius concepts to block matrix frame-
works while preserving fundamental properties and enabling new theoretical develop-
ments. We establish various properties and inequalities connecting these novel defini-
tions to established matrix parameters, thereby generalizing well-known classical results
to the block matrix setting.

The key insights of our approach lie in recognizing that while eigenvalue analysis
provides spectral information, the numerical radius captures geometric properties of the
field of values that are particularly relevant for non-normal matrices and complex block
structures. This geometric perspective complements traditional spectral methods and
provides new tools for analyzing matrix behavior in applications ranging from quantum
mechanics to control theory.

Recent advances in partial matrix theory have provided additional tools for analyzing
block matrix structures. The concept of partial spectral radius and partial matrix norms



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 4 of 15

introduced in [14] offers complementary perspectives that parallel our numerical radius
approach. Furthermore, the generalized p-numerical radius framework developed in [15]
extends classical numerical radius concepts to more general operator settings, providing
theoretical foundations that motivate our block matrix constructions.

2. Main Results and Fundamental Properties

We introduce our primary definition concerning partial matrix generalized numerical
radius.

Definition 1. Let A = [Ai,j ] ∈ Mm(Mn). We define w
(1)
N (A) = [wN (Gl,k)]

n
l,k=1 and

w
(2)
N (A) = [wN (Ai,j)]

m
i,j=1.

By specializing Definition 1 with N(A) = ‖A‖, we obtain the partial matrix of
numerical radius.

Definition 2. Let A = [Ai,j ] ∈ Mm(Mn). We define w(1)(A) = [w(Gl,k)]
n
l,k=1 and

w(2)(A) = [w(Ai,j)]
m
i,j=1.

These definitions extend the classical numerical radius to block matrix structures in
a natural way. The first partial matrix w(1)(A) captures the numerical radius properties
when we view the matrix through its column-block structure, while the second partial
matrix w(2)(A) emphasizes the row-block perspective.

The partial numerical radius constructions introduced here naturally extend to the
p-numerical radius setting studied in [15]. While [15] establishes p-numerical radius
inequalities for general operators on Hilbert spaces, our framework specializes these
concepts to block matrix structures where each block admits finite-dimensional matrix
representation. The relationship between our partial numerical radius matrices and
the partial spectral radius constructions in cite14 reveals deeper structural connections
between geometric and spectral properties of block matrices, suggesting directions for
unified treatments in future work.

Remark 1. For any matrix A = [Ai,j ] ∈ Mm(Mn), the following properties hold:

1. w
(1)
N (A) ∈ Mn(C) and w

(2)
N (A) ∈ Mm(C).

2. All entries of w(1)
N (A) and w

(2)
N (A) are positive.

3. The duality relations w
(2)
N (A) = w

(1)
N (Ã) and w

(1)
N (A) = w

(2)
N (Ã) are satisfied.

Lemma 1. For any matrix A = [Ai,j ] ∈ Mm(Mn), we have α̃A = αÃ.

Proof. This follows directly from the definition of the tilde transformation and scalar
multiplication. For αA = [αAi,j ], we have α̃A = [[αai,jl,k]

m
i,j=1]

n
l,k=1 = α[[ai,jl,k]

m
i,j=1]

n
l,k=1 =

αÃ.

The following theorem establishes the homogeneity property, which is fundamental
for any meaningful extension of the numerical radius concept.



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 5 of 15

Theorem 1. Given A = [Ai,j ] ∈ Mm(Mn), the homogeneity properties w
(1)
N (αA) =

|α|w(1)
N (A) and w

(2)
N (αA) = |α|w(2)

N (A) hold.

Proof. Consider A = [Ai,j ] ∈ Mm(Mn). Then

w
(2)
N (αA) = [wN (αAi,j)] = [|α|wN (Ai,j)] = |α|[wN (Ai,j)] = |α|w(2)

N (A).

Similarly, we can establish

w
(1)
N (αA) = w

(2)
N (α̃A) = w

(2)
N (αÃ) (by Lemma 1) = |α|w(2)

N (Ã) = |α|w(1)
N (A).

Theorem 2. For A = [Ai,j ] ∈ Mm(Mn), if either w
(2)
N (A) = 0 or w

(1)
N (A) = 0, then

necessarily A = 0.

Proof. Suppose w
(2)
N (A) = [wN (Ai,j)] = 0. This implies wN (Ai,j) = 0 for all indices

i, j = 1, 2, . . . ,m. Since wN (A) functions as a matrix norm, we conclude Ai,j = 0 for all
i, j = 1, 2, . . . ,m, yielding A = 0.

Similarly, if w
(1)
N (A) = [wN (Gl,k)] = 0, then wN (Gl,k) = 0 for all indices l, k =

1, 2, . . . , n. Again using the norm property of wN (A), we get Gl,k = 0 for all l, k =
1, 2, . . . , n, which means à = 0. Since A and à are unitarily similar, we conclude A = 0.

We introduce the concept of partial normality for block matrices, which will prove
crucial for establishing several key results.

Definition 3. A matrix A = [Ai,j ] ∈ Mm(Mn) is called partially normal when each
block Ai,j is normal for all 1 ≤ i, j ≤ m.

The concept of partial normality provides a natural generalization of matrix nor-
mality to the block setting. This property allows us to extend many classical results
about normal matrices to the block matrix framework while maintaining their essential
characteristics.

Theorem 3. For a partially normal matrix A = [Ai,j ] ∈ Mm(Mn), we have w(2)(A◦h) =
(w(2)(A))◦h.

Proof. Given A = [Ai,j ] ∈ Mm(Mn) where each Ai,j is normal, we use the property
that for normal matrices w(Bh) = wh(B) for h > 0. Therefore:

w(2)(A◦h) = [w(Ah
i,j)]

m
i,j=1 = [wh(Ai,j)]

m
i,j=1 = ([w(Ai,j)]

m
i,j=1)

◦h = (w(2)(A))◦h.

This result extends the well-known power inequality w(Ak) = (w(A))k for normal
matrices A ∈ Mm(C) and k ∈ (0,∞) to the block matrix context using Hadamard
operations.



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3. Unitary Invariance and Structural Properties

The behavior of partial numerical radius under various transformations provides
important insights into the geometric structure of block matrices.

Theorem 4. Consider matrices A = [Ai,j ] ∈ Mm(Mn), U = [Ui,j ] ∈ Mm(Mn), and
V = [Vi,j ] ∈ Mm(Mn) where each Ui,j and Vi,j is unitary for 1 ≤ i, j ≤ m. Then
w

(2)
N (A) = w

(2)
N (U ◦A ◦ V ) when N(·) is a unitarily invariant norm.

Proof. By the unitary invariance of the generalized numerical radius:

w
(2)
N (U ◦A ◦ V ) = [wN (Ui,jAi,jVi,j)]

m
i,j=1 = [wN (Ai,j)]

m
i,j=1 = w

(2)
N (A).

This theorem establishes that our partial numerical radius constructions respect the
fundamental unitary invariance property that is central to numerical radius theory.

Corollary 1. Under the conditions of Theorem 4, setting N(·) = ‖ · ‖ yields w(2)(A) =
w(2)(U ◦A ◦ V ).

Block unitary transformations provide a natural framework for analyzing the struc-
ture of partial numerical radius.

Theorem 5. Let A = [Aij ] ∈ Mm(Mn) and let P = diag(U1,1, U2,2, . . . , Um,m) be a
block diagonal unitary matrix where Uk,k ∈ Mn(C). Then:

w
(2)
N (PAP ∗) = w

(2)
N (A),

where N(·) is a unitarily invariant norm.

Proof. Since P is block diagonal, (PAP ∗)i,j = Ui,iAi,jU
∗
j,j for all i, j. By the unitary

invariance of the generalized numerical radius:

w
(2)
N (PAP ∗) = [wN (Ui,iAi,jU

∗
j,j)]

m
i,j=1 = [wN (Ai,j)]

m
i,j=1 = w

(2)
N (A).

This establishes that the partial numerical radius remains invariant under block uni-
tary similarity transformations, extending the fundamental unitary invariance property
from individual matrices to block matrix structures.

4. Block Diagonal Matrices and Triangular Structures

Block diagonal matrices represent one of the most fundamental structured matrix
classes and provide important insights into more complex block structures.

Theorem 6. Consider a block diagonal matrix A = diag(A1,1, A2,2, . . . , Am,m) ∈ Mm(Mn).
Then:

w
(2)
N (A) = diag(wN (A1,1), wN (A2,2), . . . , wN (Am,m)),

where N(·) is a unitarily invariant norm.



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 7 of 15

Proof. For a block diagonal matrix A, we have Ai,j = 0 whenever i 6= j. Therefore:

w
(2)
N (A) = [wN (Ai,j)]

m
i,j=1 (8)

= diag(wN (A1,1), wN (A2,2), . . . , wN (Am,m)) (9)

since wN (0) = 0 for all off-diagonal blocks.

Corollary 2. For block diagonal A = diag(A1,1, A2,2, . . . , Am,m) ∈ Mm(Mn):

‖w(2)(A)‖ = max{w(Ai,i) : 1 ≤ i ≤ m}

Upper triangular block matrices exhibit rich structural properties that influence their
partial numerical radius behavior.

Theorem 7. For upper block triangular A = [Aij ] ∈ Mm(Mn):

max{w(Aii) : 1 ≤ i ≤ m} ≤ w(A) ≤ max

∑
j≥i

w(Aij) : 1 ≤ i ≤ m


Proof. For the lower bound: Let xi be a unit vector achieving w(Aii) for any diagonal

block. Define y ∈ Cmn by placing xi in the i-th block position and zeros elsewhere. Then
‖y‖ = 1 and:

|〈Ay, y〉| = |〈Aiixi, xi〉| = w(Aii)

Therefore w(Aii) ≤ w(A) for all i, giving us the lower bound.
For the upper bound: For any unit vector x = (x1, . . . , xm):

|〈Ax, x〉| =

∣∣∣∣∣∣
m∑
i=1

∑
j≥i

〈Aijxj , xi〉

∣∣∣∣∣∣ (10)

≤
m∑
i=1

∑
j≥i

w(Aij)‖xj‖‖xi‖ (11)

≤ max
1≤i≤m

∑
j≥i

w(Aij)


m∑
i=1

‖xi‖2 (12)

= max
1≤i≤m

∑
j≥i

w(Aij)

 (13)

where we used
∑m

i=1 ‖xi‖2 = 1.



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 8 of 15

5. Analysis of 2× 2 Block Matrices

The 2 × 2 block case provides crucial insights that extend to larger block matrices
while remaining analytically tractable.

Theorem 8. For A =

[
A11 A12

A21 A22

]
∈ M2(Mn):

max{w(A11), w(A22)} ≤ w(A) ≤ ‖w(2)(A)‖

Proof. For the lower bound: Let x =

(
x1
0

)
where x1 maximizes w(A11). Then:

|〈Ax, x〉| = |〈A11x1, x1〉| = w(A11)
Therefore w(A) ≥ w(A11). Similarly, w(A) ≥ w(A22).

For the upper bound: For any unit vector x =

(
x1
x2

)
:

|〈Ax, x〉| ≤ w(A11)‖x1‖2 + w(A12)‖x1‖‖x2‖ (14)
+ w(A21)‖x1‖‖x2‖+ w(A22)‖x2‖2 (15)

Let y =

(
‖x1‖
‖x2‖

)
. Then: |〈Ax, x〉| ≤ 〈w(2)(A)y, y〉 ≤ ‖w(2)(A)‖

since ‖y‖ = 1.

Corollary 3. If A =

[
A11 A12

A21 A22

]
∈ M2(Mn), then the spectral norm of the partial

numerical radius matrix satisfies:

‖w(2)(A)‖ =
1

2

(
w(A11) + w(A22) +

√
(w(A11)− w(A22))2 + 4w(A12)w(A21)

)
Proof. The matrix w(2)(A) =

[
w(A11) w(A12)
w(A21) w(A22)

]
has eigenvalues given by the

quadratic formula. The largest eigenvalue provides the spectral norm.

6. Positive Semidefinite Matrices and Main Results

One of our central contributions concerns the preservation of positive semidefiniteness
under partial numerical radius transformations.

Theorem 9. Let A = [Ai,j ] ∈ Mm(Mn) be Hermitian. Then w
(2)
N (A) and w

(1)
N (A) are

symmetric and real.

Proof. Let A = [Ai,j ] ∈ Mm(Mn) be Hermitian (i.e., A = A∗). By Theorem 17, we
have w

(2)
N (A) = w

(2)
N (A∗) = (w

(2)
N (A))T = (w

(2)
N (A))∗, where the third equation follows

because the entries of w(2)
N (A) are positive real numbers. Therefore w

(2)
N (A) is Hermitian.



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 9 of 15

Similarly, w(1)
N (A) = w

(1)
N (A∗) = (w

(1)
N (A))T = (w

(1)
N (A))∗, so w

(1)
N (A) is Hermitian.

Our main theoretical contribution concerns the behavior of partial numerical radius
for positive semidefinite matrices.

Theorem 10. If A = [Ai,j ] ∈ M2(Mn) is positive semidefinite, then w(2)(A) is positive
semidefinite.

Proof. Let A =

[
A1,1 A1,2

A∗
1,2 A2,2

]
be positive semidefinite. Then for any z, y ∈ Cn:

|〈A1,2z, y〉|2 ≤ 〈A1,1z, z〉〈A2,2y, y〉 (16)

By the definition of numerical radius: w(A1,2) = max‖x‖=1 |〈A1,2x, x〉|
Setting z = y = v where v achieves the maximum in equation (16): |〈A1,2v, v〉|2 ≤

〈A1,1v, v〉〈A2,2v, v〉 ≤ w(A1,1)w(A2,2)
Hence

w2(A1,2) ≤ w(A1,1)w(A2,2) (17)

We have w(2)(A) =

[
w(A1,1) w(A1,2)
w(A1,2) w(A2,2)

]
.

By Theorem 9, w(2)(A) is Hermitian and has positive diagonal entries. For a 2 × 2
Hermitian matrix, positive semidefiniteness is equivalent to non-negative diagonal en-
tries and a non-negative determinant. The determinant of w(2)(A) is: det(w(2)(A)) =
w(A1,1)w(A2,2)− w2(A1,2) ≥ 0 by equation (17). Therefore w(2)(A) is positive semidef-
inite.

Theorem 11. If A = [Ai,j ] ∈ Mm(M2) is positive semidefinite, then w(1)(A) is positive
semidefinite.

Proof. Let A = [Ai,j ]
m
i,j=1 be positive semidefinite. Then à =

[
G1,1 G1,2

G∗
1,2 G2,2

]
is

positive semidefinite since A and à are unitarily similar. By Theorem 10, w(2)(Ã) is
positive semidefinite. By Remark 1, w(1)(A) = w(2)(Ã) is positive semidefinite.

The following example demonstrates that positive semidefiniteness preservation fails
for dimensions beyond 2× 2.

Example 1. Consider the 3× 3 matrix:

A =

 2 −1.9 0.1
−1.9 2 0
0.1 0 2


The eigenvalues of A are approximately:

λ1 ≈ 3.9045, λ2 ≈ 0.0955, λ3 = 2.0000



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 10 of 15

Since all eigenvalues are positive, A is positive semidefinite.

However, for w(2)(A) = w(1)(A) =

 2 1.9 0.1
1.9 2 0
0.1 0 2

 (using the standard numerical

radius w(·)), the eigenvalues are approximately: λ1 ≈ 4.0045, λ2 ≈ −0.0045, λ3 = 2.0000
Since λ2 < 0, w(2)(A) is not positive semidefinite, showing that the property fails for

n ≥ 3.

7. Additional Properties and Relationships

Theorem 12. If A = [Ai,j ] ∈ Mm(Mn) is partially normal, then r(2)(A) = ‖A‖(2) =
w(2)(A).

Proof. Since A is partially normal, Ai,j is normal for all 1 ≤ i, j ≤ m. By relation
(3), r(Ai,j) = ‖Ai,j‖ = w(Ai,j) for 1 ≤ i, j ≤ m. Therefore r(2)(A) = w(2)(A) = ‖A‖(2).

Theorem 13. Let A = [Ai,j ] ∈ M2(Mn) be positive semidefinite. Then:

2|w(A12)| ≤ sj

[
w(A11) w(A12)
w(A12) w(A22)

]
for j = 1, 2 (18)

|w(A12)| ≤ max{w(A11), w(A22)} (19)

where sj denotes the j-th singular value.

Proof. For a 2×2 positive semidefinite matrix
[
a b
b c

]
, we have the classical inequality

|b| ≤ 1
2sj for the singular values sj (see [2]). Applying this to our block numerical

radius matrix and using the fact that for positive semidefinite matrices Aii, we have
‖Aii‖ = w(Aii), yields the desired inequalities.

Definition 4. If A = [Ai,j ] ∈ Mm(Mn), then  = [A∗
i,j ]

m
i,j=1.

Remark 2. If A = [Ai,j ] ∈ Mm(Mn) is partially normal, then Ai,jA
∗
i,j = A∗

i,jAi,j

for i, j = 1, 2, . . . ,m. Therefore  ◦ A = A ◦ Â. Conversely, if  ◦ A = A ◦ Â, then
Ai,jA

∗
i,j = A∗

i,jAi,j, so A is partially normal.

Theorem 14. If A = [Ai,j ] ∈ Mm(Mn), then

w(A) ≤ min{w(‖A‖(1)), w(‖A‖(2))}. (20)

Proof. Since A and à are unitarily similar, w(A) = w(Ã). By using inequality (7)
from [12], we have w(A) ≤ w(‖A‖(2)) and w(A) = w(Ã) ≤ w(‖A‖(1)). Therefore,
w(A) ≤ min{w(‖A‖(1)), w(‖A‖(2))}.



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Theorem 15. For A = [Ai,j ] ∈ Mm(Mn), we have:

tr(w(1)(A)) ≤ tr(‖A‖(1)) (21)
tr(w(2)(A)) ≤ tr(‖A‖(2)) (22)

Proof. By definition of partial numerical radius, w(1)(A) = [w(Gl,k)]
n
l,k=1. For any

matrix block, w(Gl,k) ≤ ‖Gl,k‖. Taking the trace:

tr(w(1)(A)) =

n∑
l=1

w(Gl,l) (23)

≤
n∑

l=1

‖Gl,l‖ (24)

= tr(‖A‖(1)) (25)

The second inequality follows by an analogous argument.

Theorem 16. For A ∈ Mm(Mn), we have:

tr(w(1)(A)) ≤ 1

2
tr(‖A+A∗‖(1))

Proof. For each diagonal block Gll, we have w(Gll) ≤ 1
2‖Gll + G∗

ll‖ by the classical
numerical radius inequality. Summing over all diagonal blocks gives the result.

Theorem 17. Let A = [Ai,j ] ∈ Mm(Mn) and N be any self-adjoint norm. Then:

1. w
(2)
N (Aτ ) = (w

(2)
N (A))T .

2. w
(1)
N (A∗) = (w

(1)
N (A))T and w

(2)
N (A∗) = (w

(2)
N (A))T .

3. w
(1)
N (Â) = w

(1)
N (A) and w

(2)
N (Â) = w

(2)
N (A).

Proof.

1. w
(2)
N (Aτ ) = [wN (Aj,i)] = [wN (Ai,j)]

T = (w
(2)
N (A))T .

2. Since A∗ = [A∗
j,i], we have:

w
(1)
N (A∗) = w

(2)
N (Ã∗) = [wN (G∗

k,l)] = [wN (Gk,l)] (26)

= [wN (Gl,k)]
T = (w

(1)
N (A))T (27)

Similarly, w(2)
N (A∗) = [wN (A∗

j,i)] = [wN (Aj,i)] = [wN (Ai,j)]
T = (w

(2)
N (A))T .



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 12 of 15

3. Since  = [A∗
i,j ], we have:

w
(1)
N (Â) = w

(2)
N (

˜̂
A) = [wN (G∗

l,k)] (28)

= [wN (Gl,k)] = w
(1)
N (A) (29)

Similarly, w(2)
N (Â) = [wN (A∗

i,j)] = [wN (Ai,j)] = w
(2)
N (A).

8. Enhanced Theoretical Results and Applications

8.1. Characterization of Positive Semidefinite Block Matrices

Theorem 18. For A = [Ai,j ] ∈ M2(Mn), the following statements are equivalent:

(a) A is positive semidefinite.

(b) A11, A22 ≥ 0 and w2(A12) ≤ w(A11)w(A22).

(c) w(2)(A) is positive semidefinite.

Proof. (a) ⇒ (b): If A is positive semidefinite, then clearly A11, A22 ≥ 0. The
inequality follows from the proof of Theorem 10.

(b) ⇒ (c): This follows directly from the proof of Theorem 10.

(c) ⇒ (a): Suppose w(2)(A) is positive semidefinite. For any vector v =

(
v1
v2

)
:

〈Av, v〉 ≥ w(A11)‖v1‖2 − 2w(A12)‖v1‖‖v2‖+ w(A22)‖v2‖2 (30)

Since w2(A12) ≤ w(A11)w(A22), this quadratic form is non-negative, proving A ≥ 0.

Example 2 (Necessity of Conditions). Consider A =

[
1 2
2 1

]
⊗ In. Here w2(A12) = 4 >

1 = w(A11)w(A22), so condition (b) fails and A is not positive semidefinite.

8.2. Matrix Function Analysis

Theorem 19. For analytic function f and block diagonal A with Aii ≥ 0:

w(2)(f(A)) ≤ f(w(2)(A))

when f is operator monotone.

Proof. For block diagonal A = diag(A11, . . . , Amm), we have f(A) = diag(f(A11), . . . , f(Amm))
and w(2)(A) = diag(w(A11), . . . , w(Amm)).

Since f is operator monotone, w(f(Aii)) ≤ f(w(Aii)) for each i. Therefore:

w(2)(f(A)) = diag(w(f(A11)), . . . , w(f(Amm))) ≤ f(w(2)(A))



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 13 of 15

Example 3 (Failure of Triangle Inequality). The mapping A 7→ w(2)(A) does not satisfy
the triangle inequality. Consider:

A =

[
0 I
0 0

]
, B =

[
0 0
I 0

]

Then w(2)(A) =

[
0 1
0 0

]
, w(2)(B) =

[
0 0
1 0

]
, and w(2)(A+B) =

[
0 1
1 0

]
.

We have ‖w(2)(A+B)‖ = 1 while ‖w(2)(A)‖+ ‖w(2)(B)‖ = 1+1 = 2, but the matrix
inequality w(2)(A + B) 6≤ w(2)(A) + w(2)(B) demonstrates the failure of the triangle
inequality at the matrix level.

9. Conclusions

In this paper, we have successfully established a comprehensive theory of generalized
numerical radius for block matrix structures. Our main contributions include:

1. The introduction of partial matrices of numerical radius w(1)(A) and w(2)(A),
which provide new geometric insights into block matrix behavior.

2. The proof that positive semidefiniteness is preserved under partial numerical radius
transformations for 2× 2 block matrices, with explicit characterization conditions.

3. The establishment of fundamental properties including homogeneity, unitary in-
variance, and structural relationships for various classes of block matrices.

4. The development of bounds and inequalities connecting our new constructions to
classical matrix parameters.

These results extend classical numerical radius theory to block matrix frameworks
while preserving essential geometric and analytical properties. The positive semidefi-
niteness preservation result for 2 × 2 blocks is particularly significant, providing both
theoretical understanding and practical computational advantages.

Our work opens several avenues for future research. Natural extensions include
infinite-dimensional block operators, where connections to the p-numerical radius the-
ory [15] may yield fruitful insights. The relationship between our partial numerical
radius framework and the partial spectral radius approach developed in [14] suggests
opportunities for unified treatments combining spectral and geometric perspectives. Ad-
ditional directions include applications to matrix completion problems, connections to
quantum information theory, and extensions to more general unitarily invariant norms
beyond those considered here. The interplay between block structure and numerical ra-
dius properties suggests rich possibilities for further exploration in operator theory and
matrix analysis.

The limitations we have identified, such as the failure of the triangle inequality
and the restriction of positive semidefiniteness preservation to 2 × 2 blocks, provide



R. Al-Naimi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6813 14 of 15

important boundaries for the theory and suggest directions for refined approaches in
future investigations.

Data Availability Statement

Funding: Not applicable
Conflicts of interest/Competing interests: Not applicable
Availability of data and material: Not applicable
Code availability: Not applicable
Author’s contributions: All authors contributed equally to all parts of the paper.

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