EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5638
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

D-Stability Analysis of Structured Matrices Appearing
in First and Second Order Economy Models

Mutti-Ur Rehman1, Salah H. Alshabhi2, OM Kalthum S. K. Mohamed2,
Mustafa M. Mohammed2, Runda A. A. Bashir2, Mona Magzoub4,
Sakeena E. M. Hamed3, Nhla A. Abdalrahman3, Arafa O. Mustafa3,
Awad A. Bakery2,5,∗

1 Center of Research and Innovation, Asia International University, Yangiobod MFY, G‘ijduvon
Street, House 74, Bukhara, Uzbekistan
2 University of Jeddah, College of Science and Arts at Khulis, Department of Mathematics,
Jeddah, Saudi Arabia
3 University of Jeddah, College of Business at Khulis, Jeddah, Saudi Arabia
4 Mathematics Department, Faculty of Sciences and Arts-Alkamil, University of Jeddah,
Saudi Arabia
5 Ain Shams University, Faculty of Science, Department of Mathematics, Cairo, Abbassia,
Egypt

Abstract. TheD-stability of structured matrices has significant implications in system theory and
decision-making systems. The notation of D-stability refers to the characterization of a dynamical
model where structured stability is maintained when each eigenvalue of the system matrix remains
within a designated region, we consider it in half of right complex plane C subject to various
perturbations. The structured D-stability of time varying dynamical systems implies that the
system will remain stable under certain diagonal transformations. This concept is fundamental
in system theory and ensures the robust stability and performance even if system experiences
structural changes or parameter uncertainties. In this paper, novel results are obtained on the
computation of structured stability and structured D-stability of first order and second dynamical
models with mathematical forms

d

dt
(x(t)) = Ax,

d2

dt2
(x(t)) = A

d

dt
(x(t)) +Bx, x ∈ Rn,1,

with matrices A,B ∈ Rn,n. New results are developed with the necessary conditions for intercon-
nection among stable, structured D-stable matrices and structured singular values. The numerical
experimentation show how structured singular values (µ-values) behaves. The EigTool is used to
sketch the behavior of pseudo-spectrum.
2020 Mathematics Subject Classifications: 15A18, 15A16, 15A23
Key Words and Phrases: Dynamic stability, D-stability, structured singular values, pseudo-
spectrum

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5638

Email address: aabhassan@uj.edu.sa (Awad A. Bakery)

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



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 2 of 23

1. Introduction

The concept of stable and D-stable matrices play a diverse and a key role to an-
alyze behavior of linear, non-linear systems in economies models. The stable matrices
characterize and analyze the behavior of dynamical systems over time. The stability of
structured matrices in such models indicates that the deviation from an equilibrium point
will diminish over the period of time. This helps to model the economies system. The
stable matrices are particularly valuable in macroeconomic models and the dynamics of
controller. The structured D-stable operators provides a framework for the analysis, per-
formance and robustness of the stability. The structured D-stability play a versatile and
important role once the dynamical system is subject to some external perturbations, for
instance, perturbation to policy of adjustment or to vary consumer performance.

We address the problem of discussing structured stability and structured D-stability
of first and second order time varying dynamical models ẋ = Ax, and ẍ = Aẋ+Bx, x ∈
Rn,1, A,B ∈ Rn,n. We make use various tools from linear algebra, matrix analysis, and
system theory to derive some novel results. The stability of ẋ = Ax, x ∈ Rn,1 is considered
in the sense of Schur stability, that is, Re(λi(DA)) > 0, ∀i. The Jordan canonical form
of matrix-product DA is being used for derivation of novel results. Furthermore, new
results are established on interconnections between structured stable matrices, structured
D-stable matrices, and µ-values.

For matrices A,B ∈ Rn,n, we present novel results on D-stability of second order
dynamical system ẍ = Aẋ + Bx, x ∈ Rn,1. Here ẍ, ẋ denotes second, and first order
time-derivatives, respectively. These dynamical systems are determined by computing
and analyzing spectrum (eigenvalues) of 2n× 2n matrix(

A B
In O

)
.

The above structured matrix can be obtained by reducing second order system into a first
order system [1]. Furthermore, the stability of this matrix is about the computation of
strictly positive real part of the eigenvalues. On the other hand, the D-stability demands
the determination of the positive (in the strict sense) real part of spectrum of(

P11 O
0 P22

)(
A B
In O

)
.

The stability further means that the solution tends towards an equilibrium vector having
all zero entries [2]. The spectral properties of above 2n × 2n matrix were studied and
analyzed in [3] putting conditions on A,B. Further, it was shown that 2n × 2n matrix
have sufficient negative dominant diagonal elements which implies its stability.

The µ-value [4], a well-known mathematical technique to analyze and synthesize both
robust nature and the performance of linear systems from control. The applications of µ-
values includes the study and analysis of D-stability problem [5]. An exact determination
of µ-values is a very hard problem and for a class of block structured perturbations with real
parametric uncertainties it becomes an NP-hard problem [6]. There are various techniques



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 3 of 23

to compute structured singular values in the lower-dimensional problems, we refer [7,
8] and the references therein. Computing exact value of structured singular values (µ-
value) for large dimensional problems is hard and this motivates to develop the numerical
techniques for its approximation from above and below. In [9, 10], a scheme was developed
to numerically approximate the bounds of µ-values subject to mixed real and complex
uncertainties. The linear matrix inequalities (LMI) based techniques were developed [11]
to approximate µ-values from above and below.

In [12], some new results on the interconnections between the notations of H-stability,
D(α)-stability, a rank-1 perturbation to D-semistable matrices, and the µ-values were
presented and analyzed. A detailed review on mathematical methods to approximate µ-
values is presented in [13]. In [14], new results and a detailed analysis was presented on
spectrum and pseudo-spectrum of D-stable matrices for economic models.

Proposed technique in this article bridge the links in the sense that how we quantify
the interconnection between stable matrices, D-stable matrices with µ-values subject to
constraints. According to best of our knowledge not a single technique is developed to give
the interconnections of stability and D-stability of structured matrices across first order
and second order dynamical systems and µ-values. The proposed methodology determines
the novel results to discuss the stable, structured D-stable, and µ-values for structured
class of matrices appearing across the economy models.

The mathematical technique presented in this article is applicable to n-dimensional
complex valued, and real valued structured and unstructured matrices. The use of various
tools from linear algebra, matrix analysis and system theory allow us to develop a math-
ematical technique rather developing a complex geometrical based approach to discuss
the interconnections among structured stability, structured D-stability, and the µ-values.
The proposed mathematical methodology guarantees that all the equilibrium states of an
economy models shall remains in a stable mood subject to structured or unstructured
perturbations. Our technique further allows us to analyze both robust and performance
of economy models subject to structured or unstructured parametric uncertainties. The
economic models based on linear assumptions can benefit from our presented methodology
to study stability and D-stability and its interconnection with structured singular values
to ensure that a number of economic indicators will remain within considerable range.

We organized our paper as: Section 2 is on conditions for stability and D-stability,
where we have recalled basic notations, definitions, and results concerning our proposed
study. The novel results on structured stable matrices, and structured D-stable matrices
corresponding to first order dynamical systems are analyzed in section 3. In the section
4, we present the numeric experimentation’s and comparison on computation and approx-
imation of lower bounds to µ-values for structured matrices corresponding to dynamical
systems under consideration. Finally, the conclusion is presented in the section 5 of this
article.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 4 of 23

2. Conditions for stability and D-stability

Let ⃗̂x = (x̂1, x̂2, . . . , x̂n) denotes steady state of dynamical system, and fi,∀i, con-
tinuously differentiable functions, then the n-dimensional continuous (in time) dynamical
system may be written mathematically as

d

dt
xi(t) = fi (x1(t), x2(t), . . . , xn(t)) , ∀i = 1 : n.

Linearized version of above dynamical model is rewritten as follow:

d

dt
xiβ(t) =

∑
j

∂fi(x)

∂xj
xiβ(t),

where xiβ(t) = xi(t)− x̂i, ∀i.

Remark 1. If all the eigenvalues corresponding to coefficient matrix have strictly positive
real parts, that is, Re(λi(t)) > 0,∀t, then linear dynamical system is stable.

It is well-known that coefficient matrix is Jacobian of system

d

dt
xi(t) = fi(x1(t), x2(t), . . . xn(t)), ∀i = 1 : n,

and is presented by

Jc =


∂f1(x̂)
∂x1

∂f1(x̂)
∂xn

∂fn(x̂)
∂x1

∂fn(x̂)
∂xn

 .

If Re(λi(Jc)) > 0 ∀i, then system is stable. The continuous-time linear dynamic model

d

dt
x(t) = Ax(t), x(0) = x0,

with x(t) ∈ Rn,1 be a state variable, the initial condition is x0 ∈ Rn,1, and A ∈ Rn,n

matrix. An optimal and feasible solution to an optimization problem A ∈ Rn,n is obtain
as

A = arg min
dX

dt
− ÂX

F
,

where ∥ · ∥F denotes Frobenius matrix-norm, and the min is taken over  ∈ Rn,n, and
arg means the argument. Furthermore, a minimal-norm solution obtained [15] to the
dynamical system has the form A = X+ dX

dt , with X+, the pseudo-inverse to matrix X.

Remark 2. The dynamical system

d

dt
x(t) = Ax(t), x(0) = x0



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 5 of 23

may not be a stable dynamical system means that the solution matrix has the form

A = arg min
dX

dt
− ÂX

F
,

where min is taken over  ∈ Rn,n and arg means the argument.

For the structured stability of dynamical model (linear), the eigenvalues of A ∈ Rn,n

must lie in right hand side of the complex plane C.
The n-dimensional real and complex valued matrices are dented with A ∈ Rn,n, and

A ∈ Cn,n respectively. The given matrix A ∈ Cn,n is called a structured D-stable matrix
if matrix product DA appears to be a stable matrix for all D ∈ D̂, where D̂ denotes set
of positive diagonal structured matrices.

Remark 3. The matrix products DA, and AD are similar which implies that λi(DA) ∀i
are the eigenvalues of AD, because DA = D(AD)D−1.

Some simple observations [16] to D-stability of A ∈ Cn,n are:
1. It is a condition which imply stabilization, and is preserved under matrix multipli-

cation of positive structured diagonal matrices.
2. For a D-stable matrix A ∈ Cn,n such that det(A) ̸= 0, then
(a) The inversion A−1 is D-stable matrix.
(b) For P (permutation matrix), the matrix P TAP be a D-stable matrix.
(c) For D,E ∈ D̂, matrix-product DAE is a D-stable structured matrix.
(d) For A,A∗ is a structured D-stable matrix where ∗ is complex conjugate transpose

of a matrix.
3. For a D-stable matrix A ∈ Rn,n, the m × m principal sub-matrix is in Euclidean

closure of m×m D-stable structured matrices [17].

Remark 4. For A ∈ Rn,n, structured D-stable, a necessary condition is that each of
the principal minor of A is such that their determinants are non-negative. For complete
discussion, we refer [18, 19] and the references therein.

4. Let A ∈ Cn,n, and ∃ Ã ∈ D̂ such that for each Re(λi(ÃA)) > 0. Then, A is a
structured D-stable matrix if and only if det(A±D) ̸= 0 ∀D ∈ D̂.

A sufficient condition forA ∈ Cn,n to be aD-stable matrix is that λi(DA+A∗D) > 0 ∀i,
where D ∈ D̂. For more details, we refer [20–22] and references therein. The another
sufficient condition for A ∈ Cn,n to be structured D-stable matrix is that given matrix A
is an M -matrix. For A ∈ Rn,n to be a class of an M -matrix, all of its off-diagonal elements
must be less than or equal to zero while all of the principal minors to be strictly positive.
The structured M -matrices are a class of structured stable-matrices [23].

For a positive structured diagonal matrix D ∈ D̂ such that AD = B = (bij) satisfies
the strict inequality condition.

Re(bii) >

n∑
j=1,j ̸=i

|bij |, ∀i = 1 : n.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 6 of 23

Structured matrix A ∈ Cn,n satisfying above strict inequality condition is known as the
quasi-dominant diagonal matrix [24]. The given matrix A ∈ Rn,n is structured D-stable if
A = BD−1.

Remark 5. The matrix B = AD for D ∈ D̂ is such that Re(λi(B)) > 0,∀i. One can
prove this result by using Gersgorin’s circle theorem [21, 25].

A sufficient condition to D−stability to A ∈ Cn,n is that A is a triangular matrix and
further the real part of all elements of A is strictly positive, that is, Re(aii) > 0, ∀i = 1 : n.

Theorem 1. [16] Any condition to a class of structured matrices implying structured
stability, and remain preserve under multiplication of structured positive diagonal matrices
is a sufficient condition to structured D-stable matrices.

Following theorem [16] guarantees that not a single condition is the necessary condition
for the structured D-stability of matrices.

Theorem 2. Each of below class of structured matrices is a structured D-stable matrix.
1. The diagonally stable structured matrices are structured D-stable matrices.
2. The structured M -matrices are structured D-stable matrices.
3. The strictly diagonally dominant matrices having principal diagonal positive are D−stable
matrices.
4. The triangular matrices with main (principal) diagonal entries to be positive are struc-
tured D-stable matrices.
5. The sign-stable matrices are D−stable matrices.
6. The tri-diagonal structured P -matrices are structured D-stable.
7. The oscillatory matrices are structured D-stable matrices.
8. The Hadamard H-stable structured matrices are structured D-stable.
9. The structured P -matrices (sign symmetric) are structured D-stable matrices.

2.1. The µ-values and D-stability:

The stability of a dynamical system

ẋ = Ax, x ∈ Rn,1, A ∈ Km,n, K = R(or C),

demands that for given A, and for all ∆ ∈ ∆̂, λi(In+M∆) ̸= 0, ∀i. The notation ∆̂ denotes
a set of block-diagonal matrices,

∆̂ = {diag(δ1Ir1 , δ2Ir2 , . . . , δSIrS ; ∆1,∆2, . . . ,∆F ) : δi ∈ K ∀i = 1 : S; ∆j ∈ Kmj ,mj ∀j = 1 : F}.

The matrix problem to determine the necessary and sufficient conditions so that λi(In+
M∆) ̸= 0,∀i = 1 : n is a key problem in control engineering. These discussions leads us
to the definition of structured singular value.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 7 of 23

Definition 1. [26] The structured singular value (µ−value) of M ∈ Km,n, m = n with
respect to ∆̂ the set of block-diagonal matrices is defined as

µ∆̂(M) :=

{
0, if det(In −M∆) ̸= 0 ∀∆ ∈ ∆̂

1
min{∥∆∥2:∆∈∆̂, det(In−M∆)=0}

, else

where min is taken over ∆ ∈ ∆̂, ∥ · ∥2 represent maximum singular value, that is, σmax.

2.1.1. Properties of µ-values

The following properties of µ−values are easily proven and the proofs are available in
literature.

1. For α ∈ C, µ∆̂(αM) = |α|µ∆̂(M).
2. For an identity matrix In, µ∆̂(In) = 1.
3. For given matrices A,B, µ∆̂(AB) ≤ ∥A∥2 µ∆̂(B).

4. µ(∆) = ∥∆∥2 ,∀∆ ∈ ∆̂.

An alternative expression to µ−values can be easily follows from the definition.

Lemma 1. [26] For M ∈ Kn,n,

µ∆̂1
(M) = max ρ(∆M),

where max is taken over ∆ ∈ ∆̂1 with

∆̂1 := {∆ ∈ ∆̂ : σmax(∆) ≤ 1},

and the mathematical notation ρ(·) represents spectral radius (max of absolute value of an
eigenvalue) of a matrix.

Remark 6. The above lemma applies the continuity of µ : Cn,n → R which encompass the
continuity property of spectral radius as well as the max functions. It can be shown that
for S = 1, F = 0, r1 = n,

µ∆̂(M) = ρ(M),

where ∆̂ = {δIn : δ ∈ C}. Furthermore, if S = 0, F = 1, then

µ∆̂(M) = σmax(M).

The interconnection between structured singular values, the largest singular value, and
the spectral radius of M ∈ Kn,n is given by

ρ(M) ≤ µ∆̂(M) ≤ σmax(M).



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 8 of 23

3. New results for stability analysis, and D-stability analysis of first and
second order dynamical systems

In this particular section of the paper, we present new results for structured stability,
and structured D-stability analysis of first and second order dynamical systems. Further-
more, we establish the necessary conditions on the interconnection among the structured
stability, structured D-stability, and µ-values. We make use of various tools from linear
algebra, matrix analysis and system theory to derive new results.

3.1. Stability and D-stability of first order dynamical systems

First, we discuss and provide results on the stability of ẋ = Ax, x ∈ Rn,1, A ∈ Rn,n.

Assumption 1. The given matrix A ∈ Rn,n is a symmetric matrix, that is, AT = A.

Theorem 3 provides mathematical results on structured stability dynamical system
which means that the real part of its largest eigenvalue is strictly greater than real part
of the eigenvalues of all remaining eigenvalues.

Theorem 3. The first order dynamic model ẋ = Ax, x ∈ Rn,1 is stable if

Re(λ1(A)) > |Re(µi(A))| > 0, ∀i,

with λ1(A), the largest eigenvalue, and µi(A),∀i are the eigenvalues other than λ1(A).

Proof. The Re(λ1(A)) > 0 because λi(A) ∈ R,∀i and
∑
i
(λi(A)) = Trace(A) > 0, and

hence it follows that λ1(A) > 0. Now, we aim to show that

µi(A) ̸= λ1(A), Re(λ1(A)) > |Re(µi(A))| > 0, ∀i.

Let v⃗j be the normalized eigenvector for λ1 > µi,∀i, then,∑
i

aij v⃗j = µiv⃗j , ∀i.

We set |vj | = xj , then

0 < Re(λ1(A)) =
∑
ij

aij v⃗iv⃗j = |
∑
ij

aij v⃗iv⃗j | ≤
∑
ij

aijxixj .

Let xj be an eigenvector for λ1(A), then∑
j

aijxj = λ1xi, ∀i.

If we take xi = 0, for some i, then aij > 0,∀j. From this we have that each of xj = 0, this
is not possible, and hence each xj > 0, the non-decreasing condition of Re(λ1(A)). The
non-decreasing condition allows us to have that

Re(λ1(A)) >
∑
ij

aij |vi||vj | ≥ |
∑
ij

aijv
∗
i vj | = |µi(A)| = |Re(µi(A))| > 0.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 9 of 23

Theorem 4 shows that first order dynamical system is stable if the quadratic form
xTAx is strictly positive if and only if Re(λi), that is, real part of all the eigenvalues of a
real valued matrix A be strictly positive. Here, x ∈ Rn,1 is a non-zero vector.

Theorem 4. Let dynamical system ẋ = Ax, x ∈ Rn,1 is stable, then xTAx > 0, ∀x ∈ Rn,1

iff Re(λi(A)) > 0,∀i.

Proof. For the stability of dynamical system ẋ = Ax, x ∈ Rn,1, we aim to show that
for z = x + iy, x, y ∈ Rn,1, the quantity z∗Az > 0, z ∈ Cn,1. Now, we take (yTAx)T =
xTAy,AT = A. Furthermore,

z∗Az = (x+ iy)∗A(x+ iy) = xTAx+ yTAy + i(xTAy − yTAx) = xTAx+ yTAy > 0.

The strict inequality holds true if one of x and y be non-zero. This imply that Re(λi(A)) >
0, ∀i.

Theorem 5 shows that first order dynamical system is stable if the quadratic form
xTAx is strictly positive if and only if Re(λi), that is, real part of all the eigenvalues
of a complex valued matrix A is strictly positive. Here x ∈ Cn,1 is a non-zero vector
corresponding to the eigenvalues of A.

Theorem 5. The dynamical system ẋ = Ax, x ∈ Cn,1 is stable, then x∗Ax > 0 ⇔
Re(λi(A)) > 0, ∀i.

Proof. The quantity x∗Ax is a real valued and positive, then for all x ∈ Cn,1, x∗Ax is
real valued. Furthermore,

Re(λi(A)) = v∗i (λivi) = v∗iAvi, vi ∈ Cn,1,∀i

being unit eigenvectors corresponding to λi of A. This ensures that Re(λi(A)) > 0, ∀i. On
the other hand if we take A has only one eigenvalue to be positive, then

A = UΛU∗,

with U being a unitary matrix, U∗U = In and Λ = diag(λ1, λ2, . . . , λn). From this we
have that

x∗Ax = x∗UΛU∗x = (U∗x)∗Λ(U∗x) =
∑
i

Re(λi)|v∗i x|2 ≥ 0,

for Re(λi(A)) > 0, and for some non-zero v∗i x for a non-zero x. This complete the proof.

Theorem 6 shows structured D-stability analysis of the first order dynamical model.
For this, we aim to prove that A is structured D-stable if for each positive structured
diagonal matrix D, the matrix-product DA is structured stable, that is, the real part of
each of the eigenvalue (spectrum) of DA is strictly positive.

Theorem 6. The dynamical system ẋ = Ax, x ∈ Rn,1 is structured D-stable if DA is
structured stable matrix for each D = diag(dii), dii > 0, ∀ i = 1 : n.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 10 of 23

Proof. We aim to show that Re(λi(DA)) > 0,∀ i = 1 : n. In turn this will imply that
dynamical system ẋ = Ax, x ∈ Rn,1 is D-stable. The Jordan canonical form of DA is

D̂−1DAD̂ = J.

Let λi be an eigenvalues of DA, then

J =

J1 0

0 Jm

 ,

with

Jj =



λi 1 0 0
0 λi 1
0 0 λi

0
1

0 0 0 λi

 .

Let x = D̂y, then ẋ = Ax, x ∈ Rn,1 becomes ẏ = Jy. For y0 being an initial condition,
we have that, y(t) = eJty0. Now,

eJ̇t =

eJ1t 0

0 eJmt

 .

The Jordan blocks Jj , j = 1 : m can be written as Jj = λiIµi +Mµi , where

Mµi =



0 1 0 0
0 0 1
0 0 0

0
1

0 0 0 0

 ,

with Iµi , an identity matrix with order µi, i = 1 : m. Since, eλiIµi t = eλitIµi , which
can be rewritten as

eJit = eλiteMµi t.

Thus,

eJ̇it = eRe(λi)teImg(λi)tR,

where

R =


1 t t2

2!
tµi−1

(µi−1)

t2

2!
t
1

 .



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 11 of 23

Define βi =
∑
i
µi; β̂i = {βi−1 + t, . . . , βi}, then we have that

yk(t) = eRe(λi)teImg(λi)t
∑
q̂

Mp̃k q̂(µi, t)y◦q̂

where p̃k = k − βi−1.
Consider that µ := maxµi, i = 1 : m, then we have that

yk(t) = eRe(λi)teImg(λi)ttµ
∑
q̂

Mp̃k q̂(µi, t)t
−µy◦q̂.

Thus,

|yk(t)| = eRe(λi)ttµ|
∑
q̂

Mp̃k q̂(µi, t)t
−µy◦q̂|.

For t → ∞, |
∑̂
q

Mp̃k q̂(µi, t)t
−µy◦q̂| < ϵ, ϵ > 0. Since, Re(λi) > 0, ∀i, then we have that

e−Re(λi)t > tµ, i = 1 : m.

This further implies that yk(t) → 0 as t → ∞. Further, consider that λi◦ = Re(λi◦) >
0. The initial state ŷ0 so that ŷ◦q̂ = α > 0, ŷ◦γ = 0, γ ̸= q̂.

If Re(λi◦) > 0. Then

|yk(t)| = eRe(λi◦)t|
∑
q̂

Mp̃k q̂(µi◦, t)ŷ◦q̂| → ∞,

as t → 0. This ensure that |yk(t)| = ŷ◦k = α > 0. Thus, this concludes that Re(λi) > 0.

Theorem 7. The linear dynamical model ẋ = Ax, x ∈ Rn,1 is structured D-stable if
λj(A+ iD) ̸= 0, D = diag(dii) > 0, ∀ j = 1 : n.

Proof. Consider that A in ẋ = Ax is D-stable, means that, for all positive diagonal
matrices P , Re(λj(PA)) > 0,∀j. The matrix-product PA does not have i =

√
−1 as it’s

one of eigenvalue. Also, we have that

A+ iD = D(D−1A+ iIn),

such that
λj(D(D−1A+ iIn)) ̸= 0, ∀j,

where In being n× n identity matrix.
On the other hand, let A is not structured D-stable. This further implies that

Re(λj(PA)) ≤ 0,∀j. Then for a positive structured diagonal matrix D̂, D̂PA is not
structured stable but D̂A is stable, that is, Re(λj(D̂A)) > 0,∀j. This follows form the fact
that for 0 < t ≤ 1, and for some β > 0, i =

√
−1 is an eigenvalue of 1

β (tP + (1− t)In)D̂A,

where D = β(tP + (1− t)In)
−1.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 12 of 23

Theorem 8. The dynamical system ẋ = Ax, x ∈ Rn,1 is D-stable if

n∏
k=1

(λk(AD
−1 +DA−1)) > 0, ∀k.

Proof.

Consider the partitioned matrix

(
A −D
D A

)
. The Schur complement of the partitioned

matrix is
A+DA−1D = AD−1D +DA−1D = (AD−1 +DA−1)D.

By computing the kth eigenvalue of (A+DA−1D), we have that

n∏
k=1

(λk(A+DA−1D)) =
n∏

k=1

λk((AD
−1 +DA−1)D) ==

n∏
k=1

λk(A)
n∏

k=1

λk(AD
−1 +DA−1)

n∏
k=1

λk(D).

Since,
n∏

k=1

λk(A+DA−1D) > 0 ⇔
n∏

k=1

λk(AD
−1 +DA−1) > 0.

As λk(A) ≤ 0, λk(D) ≤ 0, ∀ k, and hence λk(AD
−1) ≤ 0, λk(DA−1) ≤ 0.

3.2. Stability and D-stability of second order dynamical systems

In [27], some close interconnections between µ-values and structured D-stability of real
squared matrices were developed. It was further shown that real-valued square matrix is
structured D-stable if and only if µ∆̂(M) < 1 for a given M ∈ Cn,n. The following theorem
gives an interconnection between µ−value and D−stability.

Theorem 9. [27] Consider M ∈ Rn,n. Then structured matrix M is structured D-stable
matrix ⇔ M is structured stable matrix and

0 ≤ µ∆̂((iIn +M)−1(iIn −M)) < 1.

An extension to strong structured D-stability condition for a given structured matrix
was derived in [27].

Theorem 10. [27] Let M ∈ Rn,n, and let µ∆̂(·) is the µ-value of a matrix. Then structured
matrix M is strongly D-stable iff Re(λi(M)) > 0, ∀i and ∃ ε > 0 so that 0 ≤ µ∆̂(M) < 1,
where

M =

(
(iIn +M)−1(iIn −M) 2i(iIn +M)−1

ε(iIn +M)−1 −ε(iIn +M)−1

)
.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 13 of 23

Lemma 2. [28] Let M =

(
M11 M12

M21 M22

)
. If M is strongly D-stable matrix, M22 and M c

22

are both strongly D-stable matrices, that is, for M22 or M c
22, ∃ δ > 0, such that, to each

Em ∈ Rm,m, ∥Em∥ < δ,

λi(In − (iIn +M c
22 + Em)−1(iIn −M c

22 − Em)Qm) ̸= 0,

for Qm ∈ ∆̂.

Remark 7. The matrix M c
22 is defined by Schur complement, that is, M c

22 = M22 −
M21M

−1
11 M12 with M11 such that λi(M11) ̸= 0 ∀i.

Remark 8. A sufficient condition [29, 30] for structured matrix M to be strong D-stable
structured matrix is that

inf σ1(D(In +M)−1(In −M)D−1) < 1,

where inf is taken over D ∈ D̂.

Theorem 11 shows that second order dynamical system is structured D-stable for

matrices A,B such that

(
A B
In O

)
is stable, and its non-negative structured singular value

is bounded by 1.

Theorem 11. The dynamical system ẍ = Aẋ+ Bx, x ∈ Rn,1 with A,B ∈ Rn,n, is struc-

tured D-stable if the matrix

(
A B
In O

)
is structured stable and

0 ≤ µ∆̂

((
A B
In O

)−2
)

< 1.

Proof.

We assume that

(
A B
In O

)
is stable matrix and aim to show that

0 ≤ µ∆̂

((
A B
In O

))
< 1.

From [31] it follows that

(
A B
In O

)
is D-stable if and only if Re

(
λi

(
A B
In O

))
> 0 ∀i and

n∏
i=1

λi

((
X −P
P X

))
̸= 0 ∀i, ∀P ∈ D̂, whereX =

(
A B
In O

)
.

Thus,
n∏

i=1

λi

((
X −P
P X

))
̸= 0 =⇒

n∏
i=1

λi(X
2 − PX−1PX) ̸= 0, ∀i.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 14 of 23

Furthermore, we have that

n∏
i=1

λi(X
2 − PX−1PX) ̸= 0 =⇒

n∏
i=1

λi(In −X−2P̂ ) ̸= 0,

where P̂ = P ∈ D̂, a positive diagonal matrix. Thus, finally we have that

n∏
i=1

λi(In −X−2P̂ ) ̸= 0 =⇒ 0 ≤ µ∆̂

((
A B
In O

)−2
)

< 1.

Lemma 3. Let dynamical system ẍ = Aẋ + Bx, x ∈ Rn,1, with A ∈ Rn,n such that

aii < 0, ∀i = 1 : n. Then dynamical system is structured D-stable if

(
A bIn
In O

)
is stable

for b < 0 and 0 ≤ µ∆̂

((
A B
In O

))
< 1.

Proof. Assume

(
A bIn
In O

)
is structured stable, and we aim to show that

0 ≤ µ∆̂

((
A bIn
In O

))
< 1.

The matrix

(
A bIn
In O

)
is D-stable ⇔ Re

(
λi

(
A bIn
In O

))
> 0, and

n∏
i=1

λi

((
X −P
P X

))
̸= 0,∀i, ∀P ∈ D̂, whereX =

(
A bIn
In O

)
.

Thus,

n∏
i=1

λi

((
X −P
P X

))
̸= 0 =⇒

n∏
i=1

λi(X
2 − PX−1PX) ̸= 0, ∀i, ∀P ∈ D̂.

Also,
n∏

i=1

λi(X
2 − PX−1PX) ̸= 0 =⇒

n∏
i=1

λi(In −X−2P̂ ) ̸= 0,

where P̂ = P ∈ D̂, a positive diagonal matrix. So, finally we have that

n∏
i=1

λi

(
In −

(
A bIn
In O

)−2

P̂

)
̸= 0 =⇒ 0 ≤ µ∆̂

((
A bIn
In O

)−2
)

< 1.

Theorem 12 shows that second order dynamical system is D-stable for matrices A,B

such that

(
aIn B
In O

)
is stable for a < 0, and the non-negative structured singular value is

bounded by 1.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 15 of 23

Theorem 12. The dynamical system ẍ = Aẋ + Bx, x ∈ Rn,1, is structured D-stable if(
aIn B
In O

)
, a < 0 is stable and 0 ≤ µ∆̂(X) < 1, where

X =

((
iIn O
O iIn

)
+

(
P11 O
O P22

)(
aIn B
In O

)
+

(
aIn B
In O

)T (
P11 O
O P22

))−1 ((
iIn O
O iIn

)
−

(
P11 O
O P22

)(
aIn B
In O

)
-

(
aIn B
In O

)
T

(
P11 O
O P22

)
.

Proof. To show that

(
aIn B
In O

)
is D-stable matrix iff

Re

(
λi

((
P11 O
O P22

)(
aIn B
In O

)
+

(
aIn B
In O

)T (
P11 O
O P22

)))
> 0, ∀i,

P ∈ D̂ a positive diagonal matrix, one can follow Theorem-1 of [31].

To prove that

(
aIn B
In O

)
is D-stable iff it’s structured singular values are non-negative

and bounded by 1, we let ∆, a block-diagonal structure as

∆ =

((
iIn O
O iIn

)
−
(
P11 O
O P22

))((
iIn O
O iIn

)
+

(
P11 O
O P22

))−1

,∆ ∈ ∆̂

Since,

λi

((
P11 O
O P22

)(
aIn B
In O

)
+

(
aIn B
In O

)T (
P11 O
O P22

))
̸= 0, ∀P ∈ D̂.

This implies that

λi

((
P11 O
O P22

)(
aIn B
In O

)
+

(
aIn B
In O

)T (
P11 O
O P22

)
+

(
iP11 O
O iP22

))
̸= 0, ∀i

iff λi(Y ) ̸= 0, ∀i, ∀∆ ∈ ∆̂, where

Y =

((
P11 O
O P22

)(
aIn B
In O

)
+

(
aIn B
In O

)T (
P11 O
O P22

)
+ i

[(
iIn O
O iIn

)
+

(
∆11 O
O ∆22

)])−1

((
iIn O
O iIn

)
−
(
∆11 O
O ∆22

))
.

Also, λi(Z) ̸= 0, ∀∆ ∈ ∆̂, where

Z =

(
iIn O
O iIn

)
+

(
P11 O
O P22

)(
aIn B
In O

)
+

(
aIn B
In O

)T (
P11 O
O P22

)
−
(
i

(
iIn O
O iIn

)
−
(
P11 O
O P22

)



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 16 of 23(
aIn B
In O

)
-

(
aIn B
In O

)
T

(
P11 O
O P22

)(
∆11 O
O ∆22

)
.

Finally, λi(W ) ̸= 0, ∀i, ∀∆ ∈ ∆̂, where

W =

(
In O
O In

)
−

((
iIn O
O iIn

)
+

(
P11 O
O P22

)(
In O
O In

)
+

(
aIn B
In O

)T (
P11 O
O P22

))−1 ((
iIn O
O iIn

)
−

(
P11 O
O P22

)(
aIn B
In O

)
-

(
aIn B
In O

)
T

(
P11 O
O P22

)(
∆11 O
O ∆22

)
.

This is necessary condition for µ−value to be less than 1. This complete the proof.

Theorem 13. The dynamical system ẍ = Aẋ+Bx, x ∈ Rn,1, with A,B ∈ Rn,m is D-stable

if

(
aIn B
In O

)
is stable and 0 ≤ µ∆̂(X) < 1, where

X =

((
iIn O
O iIn

)
+

(
aIn B
In O

))−1((
iIn O
O iIn

)
−
(
aIn B
In O

))
.

Proof.

The matrix

(
aIn B
In O

)
is D-stable iff

(
aIn B
In O

)
is stable and

λi

((
aIn B
In O

)
+ i

(
P11 O
O P22

))
̸= 0, ∀i,

for any P =

(
P11 O
O P22

)
positive diagonal matrix.

Consider that

(
aIn B
In O

)
is D-stable, that is

λi

((
aIn B
In O

)
+ i

(
P11 O
O P22

))
̸= 0, ∀i.

Let

∆ =

((
iIn O
O iIn

)
−
(
P11 O
O P22

))((
iIn O
O iIn

)
+

(
P11 O
O P22

))−1

be a diagonal matrix and ∆ ∈ ∆̂. Then,(
P11 O
O P22

)
=

((
iIn O
O iIn

)
+

(
∆11 O
O ∆22

))−1((
iIn O
O iIn

)
−
(
∆11 O
O ∆22

))
is positive diagonal matrix if ∆ ∈ ∆̂. As,

λi

((
aIn B
In O

)
+ i

(
P11 O
O P22

))
̸= 0,



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 17 of 23

in turn this imples that λi(X) ̸= 0,∀∆ ∈ ∆̂, where

X =

(
aIn B
In O

)
+ i

((
iIn O
O iIn

)
+

(
∆11 O
O ∆22

))−1((
iIn O
O iIn

)
−
(
∆11 O
O ∆22

))
.

The rank of(
aIn B
In O

)
+ i

((
iIn O
O iIn

)
+

(
∆11 O
O ∆22

))−1((
iIn O
O iIn

)
−
(
∆11 O
O ∆22

))
.

implies that λi(Y ) ̸= 0, ∀∆ ∈ ∆̂, where

Y =

(
In O
O In

)
−
((

iIn O
O iIn

)
+

(
aIn B
In O

))−1((
iIn O
O iIn

)
−
(
aIn B
In O

))(
∆11 O
O ∆22

)
.

Finally, this necessary condition guaratness that µ-value is less than 1.

3.3. Pseudo-spectrum:

The pseudo-spectrum of a matrix A is the set of which contains the spectrum of matrix
A. The important question one can raise is about the singularity of A which does not
appear as a robust in the sense that a small perturbation ϵ may vary the answer from yes
to no in a dramatic way. This helps to think that either ||A−1|| is large enough or not?
For λ, an eigenvalue of A, a much better question is to ask: Does ||(λIn − A)−1|| is large
or not? such a pattern allows following definitions and results [32] of pseudo-spectrum.

Definition 2. Let A ∈ Rn,n, ϵ > 0, a small perturbation. The ϵ-pseudo-spectrum σϵ(A)
which is set of eigenvalues (the spectrum) λ ∈ C such that

||(λIn −A)−1|| > 1

ϵ
.

Remark 4. For λ ∈ σ(A), σ(A) being as the set of eigenvalues of A, ||(λIn−A)−1|| =
∞. The second definition of pseudo-spectrum is given as follows.

Definition 3. Let A ∈ Rn,n, ϵ > 0, a small perturbation. The ϵ-pseudospectrum σϵ(A)
which is set of eigenvalues (the spectrum) λ ∈ C such that

λ ∈ σ(A+ E),

to some E having ||E|| < ϵ.

The third characterization of pseudo-spectrum is given as bellow.

Definition 4. Let A ∈ Rn,n, ϵ > 0, a small perturbation. The ϵ-pseudospectrum σϵ(A)
which is the of eigenvalues (the spectrum) λ ∈ C such that

||(λIn −A)v|| < ϵ

for some v ∈ Cn,1 having ||v|| = 1.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 18 of 23

4. Numerical Experimentation

We present a detailed comparison on the approximation of the bounds (from below) to
µ-values. We consider some well-known algorithm for the approximation of µ-values: The
mussv function mussv, Power Algorithm (PA) [33], Gain Based Algorithm (GBA) [34],
Poles migration Algorithm (PMA) [35], Non-linear optimization Algorithm (NLA) [36],
and Low-rank ODE’s based algorithm (LRA) given by first author [37]. The structured
matrices are taken from various economics models. We make use of EigTool [38] to compute
and display pseudo-spectrum and eigenvalues.

The function mussv is being freely available in the Matlab Control ToolBox. This tool
provides results on the numerical computation of bounds of µ-values. The proposed math-
ematical methodology give results on the numerics of bounds (from below) of µ-values of
structured matrices from various economic models. The reasons for our results closer to
the one obtained by existing methodologies are:
(i) Our mathematical methodology uses an approximation of singular values rather than
computing eigenvalues for the structured matrices.
(ii) The approximation to singular values is obtained with singular value decomposition
tool. The Matlab routine svd computes the singular values.
(iii) The approximation to largest singular value of structured matrices in the sense of
low-rank provides the sharper bounds (from below) to µ-values. The computational cost
is not high as compared with exiting techniques.
We give a number of numerical examples for economy models on the numerical approx-
imation of bounds of µ-values. The obtained results on the computation of µ-values
demonstrates the effectiveness of proposed methodology and existing techniques.

Example 1. We consider a first order dynamical model ẋ = Ax, with the coefficient
matrix A, a 4-dimensional real valued matrix taken from [39].

A =


−8 2 2 1
0 −1 1 0
0 2 −2 0
1 1 1 −5

 .

The comparison on approximation to the bounds from below to µ-values is given in the
following Table 1.

The approximated µ-values

mussv PA GBA PMA NLA LRA

8.6448 8.6421 8.6430 8.6442 8.6439 8.6448

Example 2. We consider a first order dynamical system ẋ = Ax, with the coefficient
(D-stable) matrix A, a 3-dimensional real valued matrix taken from [39].

A =

−1 −1 0
1 −2 1
0 2 −1

 .



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 19 of 23

The comparison on approximation to the lower bounds from below to µ-values is given in
the following Table 2.

The approximated µ-values

mussv PA GBA PMA NLA LRA

3.3181 3.3145 3.3167 3.3177 3.3120 3.3180

Example 3. We consider a second order dynamical system ẍ = Aẋ+ Bx, with A,B
taken from [3].

A =

−1 1 −1
8 −3 2
2 2 −2

 , B =

−1 0 0
0 −0.1 0
0 0 −0.1

 .

The matrix M is:

M =

[
A B
I3 0

]
=



−1 1 −1 0 0 0
8 −3 2 0 −0.1 0
2 2 −2 0 0 −0.1
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0

 .

The comparison on approximation to bounds from below to µ-values is given in the fol-
lowing Table 3.

The approximated µ-values of matrix M

mussv PA GBA PMA NLA LRA

8.9841 8.4940 8.2361 8.2391 8.9821 9.9838

Example 4. We consider a second order dynamical system ẍ = Aẋ+ Bx, with A,B
taken from [3].

A =

−1 0.9 0.2
0.6 −1.5 0.9
1 1 −2

 , B =

−2 0 0
0 −1 0
0 0 −2

 .

The matrix M is:

N =

[
A B
I3 0

]
=



−1 0.9 0.2 −2 0 0
0.6 −1.5 0.9 0 −1 0
1 1 −2 0 0 −2
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0

 .

The comparison on approximation to bounds from below to µ-values is given in the fol-
lowing Table 4.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 20 of 23

The approximated µ-values of matrix M

mussv PA GBA PMA NLA LRA

3.4352 3.4189 3.4310 3.4329 3.4155 3.4350

5. Conclusion

This article considers a mathematical problem related with an interconnections among
the notations of structured stability, structured D-stability analysis, and µ-values. The
novel mathematical results are developed to link the bridge between stability, structured
D-stability analysis, and µ-values. The numerical experimentation show the dynamic of
the spectrum of structured matrices. The Matlab EigTool is used for the computation of
pseud-spectrum of structured matrices across first, and second order dynamical models.
The advantages and limitations of the proposed mathematical work are listed below.

Advantages of proposed work: The proposed mathematical technique provides
an interesting interconnections between stability, D-stability of structured matrices and
their structured singular values. The new results on structured stability, and structured
D-stability may be applicable for robust analysis of nonlinear models which are subject
to Lyapunov functions. The results on D-stability ensure the stable equilibrium points
subject to various different constraints. The economic models involves structured or un-
structured uncertainties among the tax policies, the rates of interest. The study of µ-values
enable to identify the amount of structured or unstructured uncertainties affecting overall
economy while allowing policymakers to design more robust policies.
Limitations of proposed work: The D-stability, and numerical computation of µ-
values has limitations in the sense that the dynamic models subject to consideration are
most often time linear or linear time varying in their nature. The application of stability,
and D-stability the nonlinear systems may be overly simplistic. In general, the results for
stability, and D-stability may not capture the true dynamics of the economy models. The
µ-analysis is not cheap and is very costly. On the other hand, an exact determination
of µ-values is infect an NP-hard problem. The economics models most often involves a
large amount of parameters and hence it demands very high computational requirements
for the analysis of robustness and performance. This leads the practical difficulties and
complexities to scale µ-analysis to large dimensional macroeconomic models.

Acknowledgements

The authors thank the anonymous referees for their crucial and valuable comments
and suggestions, which are essential for improving the manuscript.

Conflict of Interest Statement

The authors declare that they have no conflict of interest.



M.U.R Rehman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5638 21 of 23

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and
approved the final manuscript.

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