EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 7057
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Eigenvalue Analysis and Simultaneous Nilpotence in
Intuitionistic Fuzzy Matrices

Sarah Aljohani1, Riyaz Ahmad Padder2,∗, Pratiksha Devshali3
1 Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586,
Saudi Arabia
2 Department of Mathematics, School of Chemical Engineering and Physical Sciences,
Lovely Professional University, Jalandhar, Punjab, India
3 Symbiosis Institute of Technology Pune, Symbiosis International (Deemed) University,
Pune, India

Abstract. Nilpotent intuitionistic fuzzy matrices are an important tool for analysis of intuitionis-
tic fuzzy matrices. We first examine different nilpotent conditions of such matrices in terms of their
eigenvalues in this research work. The notion of nilpotence is generalized to propose simultaneous
nilpotence for a finite set of intuitionistic fuzzy matrices. Simultaneous nilpotence is the situation
in which an infinite product of a finite number of intuitionistic fuzzy matrices approaches the zero
matrix. The basic properties of this extended concept are also formulated.
2020 Mathematics Subject Classifications: 03E72, 15B15
Key Words and Phrases: Eigenvalues, simultaneous nilpotence, directed graph

1. Introduction

Uncertainty is a key component of most areas of everyday life. Real-world issues in
medicine, engineering, industry, and economics are frequently characterized by uncertain
or imprecise data. Mathematical methods are not always at hand to overcome such diffi-
culties, which prompted Zadeh [1] to advance the idea of fuzzy set theory a revolutionary
instrument for modeling and analyzing uncertainty when classical methods were insuffi-
cient. Since then, fuzzy theory and its extensions have made enormous contributions to
a variety of mathematical applications, providing successful approaches to manifold real-
life problems with uncertainty. To address uncertainty problem, several extensions and
variations of fuzzy set theory have been developed, including vague sets, rough sets, intu-
itionistic fuzzy sets, soft sets, and so on. A Fuzzy Matrix (FM) is a matrix whose entries
are in the closed interval [0, 1]. FMs were introduced by Kim and Roush [2], and they

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7057

Email addresses: riyaz.28709@lpu.co.in (R. A. Padder),
sjohani@psu.edu.sa (S. Aljohani), pratiksha.bhavsar@sitpune.edu.in (P. Devshali)

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



S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 2 of 11

have been widely studied ever since because of their universality of application in science
and engineering and, in particular, in solving problems with uncertainty [3]. Meenakshi
[4] investageted minus ordering, space ordering, and the Schur complement of FMs and
block FMs. Ran and Liu [5], Buckley [6], and Gregory et al. [7] showed that, under the
max–min operation, an FM converges to an idempotent matrix or else has finite periodic
oscillation. Hashimoto [8] investigated the convergence property of powers of fuzzy tran-
sitive matrices. In recent years, numerous researchers have further developed the theory
of FMs [9]. While FMs contain membership values alone, Intuitionistic Fuzzy Matrices
(IFMs) consists membership and non-membership values and are a more comprehensive
framework for uncertainty modeling. IFMs were first introduced by Khan et al. [10],
and some of their properties have been studied further in [11]. Convergence of max–min
powers of IFMs was researched by Bhowmik and Pal [12], whereas mean powers of con-
vergence were examined by Pradhan and Pal [13]. Lur et al. [14] analyzed the behavior
of convergence of IFM powers. Simplify eigenvalue analysis and characterize simultane-
ous nilpotence conditions in intuitionistic fuzzy matrices through polynomial expansions
and kernel methods are given by [15–17]. Various researchers [18–29] have since further
studied IFMs and acquired useful results that have continued to enhance their use in
practical uncertainty problems. Xin [30] presented the concept of controllable FMs, while
subsequently [31] examined the convergence of their powers and established results about
nilpotent FMs. Upon this basis, this paper examines nilpotent IFM properties with respect
to their eigenvalues using digraph, thus further developing theory regarding IFMs.

Research gap

While many aspects of IFMs have been explored, the eigenvalue properties of nilpo-
tent intuitionistic fuzzy matrices have received little attention. Similarly, the concept of
simultaneous nilpotence where an infinite product of matrices converges to a zero matrix
has not been systematically studied. This oversight regarding the relationship between
eigenvalues and simultaneous nilpotence creates a gap in the theoretical understanding and
potential applications of intuitionistic fuzzy matrices, which this study seeks to address.

2. Preliminaries

Definition 1. [32] An Intuitionistic Fuzzy Set (IFS) A in X is of the form
A = {〈x, µA(x), uA(x)〉/x ∈ X}, where: µA : X → [0, 1] and νA : X → [0, 1] denotes
membership and non-membership function of the member x ∈ X, for all x ∈ X : 0 ≤
muA(x) + νA(x) ≤ 1.

Briefly we express 〈x, x′〉 as an IF member with x + x′ ≤ 1. For 〈x, x′〉, 〈y, y′〉 ∈ IFS,
For comparable members, the operation 〈x, x′〉 ↽ 〈y, y′〉 is expressed as

〈
x, x′

〉
↽
〈
y, y′

〉
=

{
〈x, x′〉 if 〈x, x′〉 > 〈y, y′〉,
〈0, 1〉 if 〈x, x′〉 ≤ 〈y, y′〉.



S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 3 of 11

Definition 2. [10] An IFM is represented by A = (〈(xi, yj), µA(xi, yj), uA(xi, yj)〉) for
i = 1, 2.m and j = 1, 2, .n, where µA : X × Y → [0, 1] and νA : X × Y → [0, 1] have the
property 0 ≤ µA(xi, yj)+νA(xi, yj) ≤ 1. IFM is a matrix can be written as A = (

〈
aij , a

′
ij

〉
)

such that aij + a′ij ≤ 1 for all i, j.

Definition 3. [33] The determinant of intutionistic fuzzy matrix is give by

|A| =

[( ∨
σ∈Sn

a1σ(1) ∧ ... ∧ anσ(n),
∧

σ∈Sn

a′1 sigma(1) ∨ ... ∨ a′nσ(n)

)]
,

where Sn denotes all the permutations groups of the indices {1, 2, ..., n}

Definition 4. Let F =
{
A(1), A(2), ..., A(m)

}
be a finte set in Fn. The IFMs

{
A(1), A(2), ..., A(m)

}
are said to be simultaneously nilpotent if Fp = {〈0, 1〉} for some p ∈ N .

In this paper, the following definition and results are used.
Let Q = [

〈
qij , q

′
ij

〉
] and S = [

〈
sij , s

′
ij

〉
] be IFMs of order n with elements in [0, 1]× [0, 1]

Q ∨ S = (
〈
qij ∨ sij , q

′
ij ∧ s′ij

〉
), where 〈x, x′〉 ∨ 〈y, y′〉 = max(〈x, x′〉 , 〈y, y′〉),

Q ∧ S = [(
〈
qij ∧ sij , q

′
ij ∨ s′ij

〉
)] where 〈x, x′〉 ∧ 〈y, y′〉 = min(〈x, x′〉 , 〈y, y′〉)

Q
c
↽ S = (

〈
qij , q

′
ij

〉
c
↽
〈
sij , s

′
ij

〉
)

Q×S = [(
〈
qi1 ∧ s1j , q

′
i1 ∨ s′1j

〉
)∨ (

〈
qi2 ∧ s2j , q

′
i2 ∨ s′2j

〉
)∨ . . .∨ (

〈
qin ∧ snj , q

′
in ∨ s′nj

〉
)],

Qk+1 = Qk ×Q, k = 1, 2, 3...

Denote Qk = [
〈
qkij , q

′k
ij

〉
], k = 2, 3, ...

(
〈
qkij , q

′k
ij

〉
=〈

n∨
j1=1

n∨
j2=1

...
n∨

jk−1=1
(qij1 ∧ qj1j2 ∧ ... ∧ qjk−1j),

n∧
j1=1

n∧
j2=1

...
n∧

jk−1=1
(q′ij1 ∨ q′j1j2 ∨ ... ∨ q′jk−1j

)

〉
Q0 = I, (In is unit matrix ) QT = [

〈
qji, q

′
ji

〉
] (the transpose)

∆Q = Q
c
↽ QT , ∇Q = Q ∧QT

Q ≤ S iff (
〈
qij , q

′
ij

〉
≤
〈
sij , s

′
ij

〉
for all i, j ∈ 1, 2, ..., n).

The IFM Q is known as nilpotent if Qn = (〈0, 1〉),
Controllable matrix T = [

〈
tij , t

′
ij

〉
] = P ×Q× P T satisfies

〈
tij , t

′
ij

〉
≥
〈
tji, t

′
ji

〉
for i > j.

3. Eigenvalue and nilpotence

Definition 5. Let (〈aij , a′ij〉) be n×n IFM, λ ∈ [0, 1] is known as eigenvalue of (〈aij , aij〉)
if (〈aij , a′ij〉)

c
↽ 〈x, x′〉 = λ

c
↽ 〈x, x′〉 for nonzero vector 〈x, x′〉 = [〈xj , x′j〉] with 〈xj , x′j〉 ∈

[0, 1], namely,∨
j=1

(〈aij , a′ij〉 ∧ 〈xj , x′j〉) = λ ∧ 〈xi, x′i〉, for all i = 1, 2, ..., n.

Consider σ(〈aij , a′ij〉) set of all eigenvalues of A and let ρ(〈aij , a′ij〉) = sup
{
λ|λ ∈ σ(〈aij , a′ij〉)

}
.



S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 4 of 11

We will prove in Theorem 2 that for any IFM 〈aij , a′ij〉, there exists a λ ∈ σ(〈aij , a′ij〉).
Thus, ρ((〈aij , a′ij〉)) is the largest eigenvalue of (〈aij , a′ij〉).

Lemma 1. Let (〈aij , a′ij〉) ∈ Fn. Then (〈aij , a′ij〉) has a zero column ⇔ 〈0, 1〉 ∈ σ((〈aij , a′ij〉))

Proof. Let ith column of (〈aij , a′ij〉) is 0. And 〈x, x′〉 = ei, then 〈x, x′〉 is an eigenvector
corresponding to the 〈0, 1〉.
Let 〈x, x′〉 = (〈xi, x′i〉) be an eigenvector corresponding to the eigenvalue 〈0, 1〉. Suppose
that 〈xi, x′i〉 6= 〈0, 1〉. Then

(〈aij , a′ij〉)
c
↽ 〈x, x′〉 =


n∨

j=1
(〈a1j , a′1j〉 ∧ 〈xj , x′j〉

...
n∨

j=1
(〈anj , a′nj〉 ∧ 〈xj , x′j〉

 = O

Thus, 〈aki, a′ki〉 ∧ 〈xi, x′i〉 = 〈0, 1〉 for all k. Since 〈xi, x′i〉 6= 〈0, 1〉,
we have 〈aki, a′ki〉 = 〈0, 1〉 for every k. Hence ith column of (〈aij , a′ij〉) is 〈0, 1〉

Lemma 2. Let 〈aij , a′ij〉 ∈ Fn. Then ρ(〈aij , a′ij〉) is 〈0, 1〉 or 〈1, 0〉.

Proof. If σ(〈aij , a′ij〉) = 〈0, 1〉, then ρ(〈aij , a′ij〉) = 〈0, 1〉. Otherwise, if ∃ 〈0, 1〉 6=
λ ∈ σ(〈aij , a′ij〉), then for nonzero eigenvector 〈x, x′〉 we have (〈aij , a′ij〉)

c
↽ 〈x, x′〉 = λ

c
↽

〈x, x′〉. For any β with λ ≤ β ≤ 〈0, 1〉, we have
〈aij , a′ij〉

c
↽ (λ

c
↽ 〈x, x′〉) = λ

c
↽ 〈x, x′〉 = β

c
↽ (λ

c
↽ 〈x, x′〉).

Hence, β ∈ σ(〈aij , a′ij〉 ⇒ ρ(〈aij , a′ij〉) = 〈1, 0〉

Lemma 3. Let A = (〈aij , a′ij〉) and B = (〈bij , b′ij〉) ∈ Fn. If (〈aij , a′ij〉) ≤ (〈bij , b′ij〉), then
ρ(〈aij , a′ij〉) ≤ ρ(〈bij , b′ij〉).

Proof. By Lemma 2 , ρ(〈aij , a′ij〉) is either 〈0, 1〉 or 〈1, 0〉. If ρ(〈aij , a′ij〉) = 〈0, 1〉, the
result is trivial. Now, we will show that ρ(〈aij , a′ij〉) = 〈1, 0〉 ⇒ ρ(〈bij , b′ij〉) = 〈1, 0〉. Let
(〈aij , a′ij〉)

c
↽ (〈x, x′〉) = (〈x, x′〉), (〈x, x′〉) 6= O. Then (〈x, x′〉) ≤ (An) = ((〈anij , a′nij 〉) c

↽
)

c
↽

e (by Lemma 2.10 [34]) , where e = [1, 1, ..., 1] and so (〈aij , a′ij〉) ≤ ((〈anij , a′nij 〉) c
↽
)

c
↽

e ≤ (Bn) = ((〈bnij , b′nij 〉) c
↽
)

c
↽ e (because (〈aij , a′ij〉) ≤ (〈bij , b′ij〉). Since 〈x, x′〉 6= O,

since ((〈bnij , b′nij 〉) c
↽
)

c
↽ e ≤ O. Let 〈y, y′〉 = ((〈bnij , b′nij 〉) c

↽
)

c
↽ e. Then (〈bij , b′ij〉)

c
↽

〈y, y′〉 = ((〈bn+1
ij , b′n+1

ij 〉) c
↽
)

c
↽ e = ((〈bnij , b′nij 〉) c

↽
)e = 〈y, y′〉 (By Lemma 2.9 [34]) and so,

ρ(〈bij , b′ij〉) = 〈1, 0〉.

Theorem 1. Let (〈aij , a′ij〉) ∈ Fn.
(1) (〈aij , a′ij〉) is nilpotent .
(2) (〈aij , a′ij〉)

p
c
↽

= O for some integer p.
(3) σ((〈aij , a′ij〉)) = {〈0, 1〉}
(4) ρ((〈aij , a′ij〉)) = 〈0, 1〉



S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 5 of 11

Proof. Let (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (1).In these sequence, (1) ⇒ (2) and (3) ⇒ (4)
are trivial. Now we will prove (2) ⇒ (3). Suppose λ ∈ σ((〈aij , a′ij〉)) is a eigenvalue which
is nonzero. Then (〈aij , a′ij〉)

c
↽ 〈x, x′〉 = λ

c
↽ 〈x, x′〉, where 〈x, x′〉 is a corresponding

eigenvector. Implies that
O = ((〈apij , a

′p
ij〉) c

↽
)

c
↽ 〈x, x′〉 = λp c

↽ 〈x, x′〉 = λ
c
↽ 〈x, x′〉 6= O,

which is not possible. Now, we will prove that (d) ⇒ (a). Assume that (〈anij , a′nij 〉) c
↽

6= O.
Then ((〈anij , anij〉) c

↽
)

c
↽ e 6= O.

Let 〈y, y′〉 = ((〈anij , a′nij 〉) c
↽
)

c
↽ e.

Then (〈aij , a′ij〉)
c
↽ 〈y, y′〉 = ((〈an+1

ij , an+1
ij 〉) c

↽
)e = ((〈anij , a′nij 〉) c

↽
)e = (〈y, y′〉), and so

ρ((〈aij , a′ij〉)) = 〈1, 0〉, which is contradiction.

Theorem 2. Let (〈aij , a′ij〉) ∈ Fn. Then the set σ((〈aij , a′ij〉)) has following properties.
(1) (〈aij , a′ij〉) is nilpotent iff σ((〈aij , a′ij〉)) = 〈0, 1〉.
(2) (〈aij , a′ij〉) is non nilpotent, every column of (〈aij , a′ij〉) is 6= 0 iff σ((〈aij , a′ij〉)) = (0, 1]
(3) (〈aij , a′ij〉) is non nilpotent, contains at least one zero column iff σ((〈aij , a′ij〉)) = [0, 1]

Proof. Condition (1) has been shown in Theorem 1. (2) By Lemma 1 and condition
(1), ρ((〈aij , a′ij〉)) = 〈1, 0〉.
Let 〈1, 0〉 is the eigenvalue of (〈aij , a′ij〉) with nonzero eigenvalue 〈x, x′〉. For 〈0, 1〉 <
〈β, β′〉 ≤ 〈1, 0〉 since,
(〈aij , a′ij〉)(〈β, β′〉〈x, x′〉) = 〈β, β′〉〈x, x′〉 = 〈β, β′〉(〈β, β′〉〈x, x′〉).
Thus [0, 1] ⊂ σ((〈aij , a′ij〉)). Every column of (〈aij , a′ij〉) is 6= 0, by Lemma 1 i.e (0, 1] =
σ((〈aij , a′ij〉)). Inverse follows from Lemma 1 and Properety (1). Proof of (3) similar to
that of (2), expect by adding that 〈0, 1〉 ∈ σ((〈aij , a′ij〉) because (〈aij , a′ij〉) had a zero
column. Thus [0, 1] = σ((〈aij , a′ij〉)).

Example 1. Let (〈aij , a′ij〉) =
(
〈0, 1〉 〈0.5, 0.5〉
〈0, 1〉 〈0, 1〉

)
,

(〈bij , b′ij〉) =
(
〈0.5, 0.5〉 〈0.5, 0.5〉
〈0, 1〉 〈0, 1〉

)
be irreflexive, antisymmetric and w-transitive IFM. Now, let

(〈cij , c′ij〉) =
(
〈0, 1〉 〈0.5, 0.5〉
〈0, 1〉 〈0.5, 0.5〉

)
.

We note that IFM (〈aij , a′ij〉) is Nilpotent. Computation shows that σ((〈aij , a′ij〉)) =
〈0, 1〉 . It also shows by impilication operator (〈bij , b′ij〉) and (〈cij , c′ij〉) are not nilpotent
IFM, Moreover, σ((〈bij , b′ij〉)) = (0, 1] and σ((〈cij , c′ij〉)) = [0, 1].

Theorem 3. If F =
{
(〈aij , a′ij〉)(1), (〈aij , a′ij〉)(2), ..., (〈aij , a′ij〉)(m)

}
⊂ Fn. Then

1. F is simultaneously nilpotent
2. Each Principal minor of (〈mij ,m

′
ij〉) is 〈0, 1〉 for all (〈mij ,m

′
ij〉) ∈

⋃
k≥1Fk

3. The digraph Γ(F) is acyclic.



S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 6 of 11

Proof. (1) ⇒ (2). let that det((〈mij ,m
′
ij〉)[α]) 6= 〈0, 1〉 for some (〈mij ,m

′
ij〉) ∈

⋃
k≥1Fk

and [α] = α1, α2, α3..., αl ⊂ 1, 2, 3, ...n. Then ∃ permutation σ on [α] such that [(〈mij ,m
′
ij〉)]αiσ(αi) 6=

〈0, 1〉 for i = 1, 2, ...l. Let (β1, β2..., βr) cycle in σ. Then
(〈mij ,m

′
ij〉)βiβi+1

6= 〈0, 1〉 for i = 1, 2, 3, ...r − 1

and [(〈mij ,m
′
ij〉)]βrβ1 6= 〈0, 1〉

Hence, [(〈mij ,m
′
ij〉)rc↽]βiβi

6= 〈0, 1〉 ∀ i = 1, 2, ..., r − 1. That is (〈mij ,m
′
ij〉)nrc↽ 6= 〈0, 1〉.

Thus, Fnq 6= {O} for some q, and so Fn 6= {O} that contradicts.
(2) ⇒ (3). Assume that Γ(F) contains a cycle γ(υβ1 , υβ2 , ..., υβk

, υβ1). Then ∃ (〈aij , a′ij〉)1, ..., (〈aij , a′ij〉)k
in F such as
[(〈aij , a′ij〉)i]βiβi+1

6= 〈0, 1〉 for i = 1, 2, ...k − 1. and
[(〈aij , a′ij〉)k]βkβ1 6= 〈0, 1〉
That is [(〈aij , a′ij〉)1

c
↽ ...

c
↽ (〈aij , a′ij〉)k]β1β1 6= 〈0, 1〉 and hence,

det(((〈aij , a′ij〉)1
c
↽ ...

c
↽ (〈aij , a′ij〉)k)[β1] = [(〈aij , a′ij〉)1

c
↽ ...

c
↽ (〈aij , a′ij〉)k]β1β1 6= 〈0, 1〉 .

(3) ⇒ (1). Let (〈aij , a′ij〉)1, (〈aij , a′ij〉)2..., (〈aij , a′ij〉)n ∈ F . For every i, j, ∃ i1 = i,
i2, ..., in, in+1 = j as
[(〈aij , a′ij〉)1

c
↽ ...

c
↽ (〈aij , a′ij〉)n]ij = [(〈aij , a′ij〉)1]i1i2

∧
...
∧
[(〈aij , a′ij〉)n]inin+1 .

Since {i1, ..., in+1} ⊂ {1, ..., n}, there are 1 ≤ r < s ≤ n + 1 so that ir = is. As Γ(F)
contains no cycles,
[(〈aij , a′ij〉)r]irir+1

∧
...
∧
[(〈aij , a′ij〉)s−1]is = 〈0, 1〉

Hence,
[(〈aij , a′ij〉)1

c
↽ ...

c
↽ (〈aij , a′ij〉)n]ij = [(〈aij , a′ij〉)1]i1i2

∧
...
∧
[(〈aij , a′ij〉)n]inin+1 = 〈0, 1〉

Therefore, Fn = {O} .
Let F =

{
(〈aij , a′ij〉)(1), (〈aij , a′ij〉)(2)...(〈aij , a′ij〉)m

}
⊂ Fn. It is clear Γ(F) consists a

directed path having length k iff there is
(〈aij , a′ij〉)i1 , (〈aij , a′ij〉)i2 , ..., (〈aij , a′ij〉)ik ∈ F so that (〈aij , a′ij〉)i1

c
↽ (〈aij , a′ij〉)i2 , ...

c
↽

Aik 6= 〈0, 1〉. Thus, develop the theorem to find the index h(F) of F , where h(F) is the
least positive integer k so that h(Fn) = {O}. And If h(F) = {A}, then h(F) is denoted
by h((〈aij , a′ij〉)).

Theorem 4. Let F =
{
(〈aij , a′ij〉)(1), (〈aij , a′ij〉)(2), ..., (〈aij , a′ij〉)(m)

}
⊂ Fn be a simulta-

neously nilpotent set. Then h(Fk) = k ≥ 2 iff the digraph Γ(F) contains directed paths
having k − 1 length, but none of the directed paths having k lengths.

Proof. Assume Γ(F) consists directed paths of k − 1 length and no directed path of
k length. By definition of adjacency matrix of Γ(F), the ijth entry of Al denotes the
number of directed paths of length l from vi to vj . Thus, if there is a directed path of
length ≤ k − 1, then At 6= 〈0, 1〉, for 1 ≤ t ≤ k − 1 and since there is no directed path of
length k, so At 6= 〈0, 1〉 then Γ(F) = k. Converse follows by following the above steps in
reverse order.



S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 7 of 11

Example 2. Let

B(1) = (〈bij,b
′
ij〉)(1) =



v1 v2 v3 v4 v5

v1 〈0, 1〉 〈0.1, 0.9〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v2 〈0, 1〉 〈0, 1〉 〈0.1, 0.9〉 〈0, 1〉 〈0, 1〉
v3 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0.1, 0.9〉 〈0, 1〉
v4 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v5 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0.1, 0.9〉



B(2) = (〈bij,b
′
ij〉)(2) =



v1 v2 v3 v4 v5

v1 〈0, 1〉 〈0, 1〉 〈0.01, 0.99〉 〈0, 1〉 〈0, 1〉
v2 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0.01, 0.99〉 〈0, 1〉
v3 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0.01, 0.99〉
v4 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v5 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉



B(3) = (〈bij,b
′
ij〉)(3) =



v1 v2 v3 v4 v5

v1 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0.01, 0.99〉 〈0, 1〉
v2 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0.01, 0.99〉
v3 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v4 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v5 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉



B(4) = (〈bij,b
′
ij〉)(4) =



v1 v2 v3 v4 v5

v1 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0.01, 0.99〉
v2 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v3 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v4 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v5 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉



B(5) = (〈bij,b
′
ij〉)(5) =



v1 v2 v3 v4 v5

v1 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v2 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v3 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v4 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉
v5 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉 〈0, 1〉





S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 8 of 11

V1

2

34

5 V

V

V

V

B
(1)

V1

2

34

5 V

V

V

V

B
(2)

V1

2

34

5 V

V

V

V

B
(3)

V1

2

34

5 V

V

V

V

B
(4)

V1

2

34

5 V

V

V

V

B
(5)

Figure 1: Simultaneous nilpotence.

Note: Zhang [35] studied the index h(F) by using E(〈aij ,a′ij〉) =
{
(i, j)|〈aij , a′ij〉 ≥ 〈0, 1〉

}
is the set of all directed edges in Γ((〈aij , a′ij〉)),
E

(1)
(〈aij ,a′ij〉)

=
{
i|(i, j) ∈ E(〈aij ,a′ij〉)

}
is the set of all initial vertices of directed edges in Γ((〈aij , a′ij〉)),
and E2

2 =
{
j|(i, j) ∈ E(〈aij ,a′ij〉)

}
is the set of all end vertices of directed edges in Γ((〈aij , a′ij〉)).

Similarly we can extend with simultaneous nilpotence.
Let F =

{
(〈aij , a′ij〉)(1), (〈aij , a′ij〉)(2), ..., (〈aij , a′ij〉)(m)

}
⊂ Fn×n be a simultaneously nilpo-

tent set .
Define EF , E

(1)
F , E

(2)
F by EF =

{
(i, j) : 〈aij , a′ij〉 6= 〈0, 1〉 for some (〈aij , a′ij〉) = [〈aij , a′ij〉] ∈ F

}
;

E
(1)
F =

{
i : 〈aij , a′ij〉 6= 〈0, 1〉 for some (〈aij , a′ij〉) = [〈aij , a′ij〉] ∈

}
;

E
(1)
F =

{
i : 〈aij , a′ij〉 6= 〈0, 1〉 for some (〈aij , a′ij〉) = [〈aij , a′ij〉] ∈

}
.

Theorem 5. Let (〈aij , a′ij〉) be IFM . Then
(1) (〈aij , a′ij〉) is nilpotent IFM.
(2) The digraph Γ((〈aij , a′ij〉)) is acyclic. (3) Each Principal minor of (〈aij , a′ij〉) is 〈0, 1〉
(4) Every Principal minor of (〈aij , a′ij〉)n is 〈0, 1〉, n = 1, 2, ..



S. Aljohani, R. A. Padder, P. Devshali / Eur. J. Pure Appl. Math, 18 (4) (2025), 7057 9 of 11

Proof. Proof follows directly follows from theorem 4.

4. Conclusion

This paper explores the issue of nilpotent intuitionistic fuzzy matrices. Since all intu-
itionistic fuzzy matrices have eigenvalues, we examined their potential sets and concluded
that they could be divided into three categories:〈0, 1〉, (0, 1], [0, 1],. It is also demonstrated
that nilpotence is specially defined by the eigenvalue 〈0, 1〉. With finite product of IFMs,
the notion of simultaneous nilpotence arises, meaning the infinite products of a finite
number of such matrices strongly converge to the zero matrix, this characteristic is fur-
ther defined using principal minors and directed graphs. The theoretical findings have
possible applications in multi-criteria decision-making, stability analysis of fuzzy dynam-
ical systems, network modeling, control theory, and intelligent decision-support systems,
where uncertainty and convergence phenomena play a dominant role. Future work could
generalize these results to higher-dimensional fuzzy structures, study computational al-
gorithms for determining simultaneous nilpotence, and examine practical applications in
medical diagnosis, supply chain management, and machine learning.

Acknowledgements

The author S. Aljohani would like to thank Prince Sultan University for paying the
APC and for the support through the TAS research lab.

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