EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5657 ISSN 1307-5543 – ejpam.com Published by New York Business Global Analysis of Stability, D-Stability, and Pseudospectra in Economic Modeling Mutti-Ur Rehman1, Sakeena E. M. Hamed3, Nidal E. Taha2, Arafa O. Mustafa3,∗, Khurshidbek Dilmurodov1, Hala S. Mahgoub3, Mona Magzoub4, Runda A. A. Bashir5, Mustafa M. Mohammed5,∗, Awad A. Bakery5,6 1 Center of Research and Innovation, Asia International University, Yangiobod MFY, G‘ijduvon Street, House 74, Bukhara, Uzbekistan 2 Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia 3 University of Jeddah, College of Business at Khulis, Jeddah, Saudi Arabia 4 Mathematics Department, Applied College at Alkamil, University of Jeddah, Saudi Arabia 5 University of Jeddah, Applied College at Khulis, Department of Mathematics, Jeddah, Saudi Arabia 6 Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt Abstract. The analysis of dynamic stability is a fundamental and an important concept in system dynamics. Its focus is on the ability of a dynamical system to return to an equilibrium state under structured perturbations. The study of dynamic stability plays critical role in various fields, for instance, engineering, control theory, and economics. The analysis on the dynamic stability mostly involves computation of the eigenvalues of a system’s state matrix. The D-stability is a particular and specialized form of dynamic stability, and its mainly focus dynamical systems subject to structured perturbations. In this paper, we present new results on dynamic stability, and D-stability of a class of linear economic model in the mathematical form yt = Ayt +Byt−1 + Cxt, with yt, a vector of the endogenous variables, xt is a vector of exogenous variables, and A,B, and C are the matrices having an appropriate dimensions. The new results are developed on both nec- essary and sufficient conditions on the interconnection between D-stable matrices and structured singular values. The numerical experimentation show the behaviour of structured singular values for matrices appearing across linear dynamic model. 2020 Mathematics Subject Classifications: 15A18, 15A16, 15A23 Key Words and Phrases: Dynamic stability, D-stability, structured singular values, pseudo- spectrum ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5657 Email addresses: mustasta@yahoo.com and mmibrahim@uj.edu.sa (Mustafa M. Mohammed) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 2 of 17 1. Introduction A vast amount of literature has been written on the problem related to the control of economics. The main concentration was given to study and analyze the static economic models. The most of the economics models are dynamic in their nature. The study on the current state of an economic system involving a vast amount of policies are being used to shift system from current status to future state while dealing with such dynamic systems. For the input-output economics, a vast amount of literature has been written in order to describe the real economics [25, 31]. The general linear dynamic systems are much easier to deal with compare to singular dynamic systems such as implicit dynamic sys- tems, generalized dynamic systems, generalized state-space dynamic systems, semi-state dynamic systems [9, 24]. A vast amount of literature has been written on regular dynamic systems and descriptor systems, we refer interested reader to see [14, 15, 42] and references therein. The linear matrix inequalities techniques were developed to study the singular dynamic system in economics, see [10, 35, 36, 43, 45]. The new results on the interconnections between dynamic behavior of macroeco- nomics, continuous-time dynamical models and relationships between dynamic stability and dominant-diagonal structure was studied and analyzed in [30]. The most of macro- economics continuous-time dynamic models appears to be non-stable. In [5, 6, 11, 37] it was shown that macro-economic continuous-time models are unstable. For general discrete-time dynamical models appearing in economics, the relationship between the dom- inant diagonals and the stability was developed by [16, 21, 26, 27, 41]. The D-stability for a class of real valued matrices to study the equilibrium in dynamic models of competitive market for the first time was studied by Arrow and McManus [2], and Enthoven and Arrow in [13]. The study of dynamic stability of tatonnement process for Walrsian model of general equilibrium attracted a major community of economists. The classical approach developed by Sanuelson describes the dynamic behaviour of the economic models. The new results on D-stability, strong D-stability and structured singular values were developed by using various tools from linear algebra, matrix analysis, and system the- ory, see [34]. In [22], the most general general relationships between performance and robustness of dynamical system, and special type of matrix stabilities, that is, D stability and diagonal stability were studied and analyzed. The results on new stability condi- tions for second-order dynamical systems were presented and analyzed. An extension to D-stability for non-square matrices which are applicable to distributed and decentralized controllability analysis was recently studied in [40]. The µ-values or structured singular values first introduced and analyzed by J. C. Doyel [12] and Safonov [38] is a mathematical technique in order to investigate and test the stability of linear dynamical systems. In general problem aiming the determination of stability in the presence of structured or un- structured uncertainties is most fundamental issue in control and has attracted researchers from almost last three decades. In this article, we present new results on stability and D-stability of linear dynamic models that appears in economics. We also present new results on necessary and sufficient M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 3 of 17 conditions on interconnections between D-stability and structured singular values of ma- trices appearing across dynamic models. The numerical experimentation shows the com- parison of structured singular values for matrices with various dimensions from dynamic models. Finally, the pseudo-spectrum of matrices across dynamic models is presented while making use of EigTool [28]. Overview of article: In Subsection 1.1, we give the preliminaries to recall important concepts, and definition to be used in this article. In Section 2, we provide new results to study dynamic stability and D-stability of economic models. We make use of various tools from linear algebra, system theory to derive these new results. In Section 3, necessary and sufficient conditions are derived for the D-stability of dynamical system. The numerical experimentation to support new results are presented in Section 4. We have made use of Eigtool to visualize the pseudo-spectrum of structured matrices across the dynamical systems. Finally, in Section 5, we conclude our paper. 1.1. Preliminaries. For M ∈ Cn,n, the largest singular value is denoted by σmax, and is a non-negative real number. The smallest singular value is denoted by σmin. The notation MT denotes the transpose of a matrix, λi(M) denotes all the eigenvalues of the matrixM . ForM > 0(M ≥ 0) means that matrix M is positive definite, and positive semi-definite, respectively. For M < 0 (M ≤ 0) means that matrix M is negative definite, and negative semi-definite, respectively. The symbol C+. Definition 1. A block diagonal matrix D is defined as D = diag(D11, D22, ···, Dnn), where D11, · · ·, Dnn, are the matrices. Definition 2. The block-diagonal structure ∆ represents the uncertainty set and is defined as ∆ =: {D = diag(D11, D22, · · ·, Dnn)}, D ∈ Rn,n. Definition 3. The set ∆+ is defined as ∆+ =: {D ∈ ∆ : Dii > 0, ∀i = 1 : n}. Definition 4. The singular values σi ∀ i = 1 : n are the non-negative numbers appearing in the diagonal matrix Σ in the singular value decomposition of a matrix M = UΣV T , with U, V being as real orthogonal matrices. Definition 5. For a given matrix M ∈ Cn,n, and an underlying set ∆, the structured singular value is defined [33] as µ∆(M) := 1 min{||∆̂|| : det(In −M∆̂) = 0, ∀∆̂ ∈ ∆} , otherwise µ∆(M) = 0 if det(In −M∆̂) ̸= 0, ∀∆̂ ∈ ∆. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 4 of 17 Note that min is taken over ∆̂ ∈ ∆, and min over an empty set is +∞. Definition 6. [4, 17]. The n-dimensional real valued matrix A ∈ Rn,n is continuous-time stable if ∃ D ∈ ∆+ such that (ATDA−D) < 0. Theorem 1. [23] The continuous-time linear system dx(t) dt = Ax(t), x(t) ∈ Rn,1 is asymp- totically stable iff Re(λi) < 0 ⇔ π 2 < ϕ < 3π 2 , ∀ i = 1 : n, with λi = |λi|eιϕi , ∀ i = 1 : n, the eigenvalues of A. Note that the matrix  ∈ Rn,n is discrete-time diagonal stable if ∃ D ∈ ∆+ such that (ÂTDÂ−D) < 0. Definition 7. [8, 44]. The n-dimensional real valued matrix A ∈ Rn,n is continuous-time D-stable if the product DA is such that Re(λi(DA)) < 0, ∀i, D ∈ ∆+, a positive diagonal matrix. Definition 8. [3]. The n-dimensional real valued matrix  ∈ Rn,n is discrete-time D- stable if the product D is such that ρ(DÂ) < 1, for all real valued diagonal and norm bounded matrices D. Theorem 2. [23] The discrete-time linear system xi+1 = Āxi, xi ∈ Rn,1, i = Z+ is asymptotically stable iff |λi| < 1 with λi, ∀ i = 1 : n, are the eigenvalues of Ā. 2. Dynamic stability and D-stability of economic models In the section, we present new results to study the dynamic stability and D-stability of a class of linear economic model given in the mathematical form yt = Ayt +Byt−1 + Cxt. Here, yt is a vector of the endogenous variables, and xt is a vector of exogenous vari- ables. A,B, and C are the matrices having an appropriate dimensions. The reduced mathematical form of the above dynamic model is given as yt = (In −A)−1Byt−1 + Ext. If the spectral radius, that is, ρ((In − A)−1B) < 1, then dynamic model is said to be dynamically stable. The following Theorem 3 show that the reduced dynamic model is dynamically stable. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 5 of 17 Theorem 3. Let A,B ∈ Rn,n. Then dynamical model yt = (In − A)−1Byt−1 + Ext is dynamically stable if ρ((In −A)−1B) < 1. Proof. We aim to prove that ρ(In −A)−1B) < 1 by using inequality ρ((In −A)−1B)) ≤ ||(In −A)−1B)||2. In order to prove our result, we assume that inequality holds true for strict inequality, that is, ρ((In −A)−1B)) < ||(In −A)−1B||2. For the matrix (In −A)−1B, there exists unitary matrices U ∈ Cm,n, V ∈ Cm,n such that (In −A)−1B = U ( σ1 0 0 T ) V H . Next, we consider σ1 and θ1 ∈ Cn,1 so that σ1 = ∥(In −A)−1Bθ1∥2 = ∥(In −A)−1B∥2, while ∥θ1∥2 = 1. Further, we let u1 = (In−A)−1Bθ1 σ1 such that ∥u1∥2 = ∥(In −A)−1Bθ1∥2 σ1 = ∥(In −A)−1Bθ1∥2 ∥(In −A)−1B∥2 = 1. Consider that U2 ∈ Cm,m−1, V2 ∈ Cn,n−1. Thus, U and V takes the form U = (u1|U2), and V = (v1|V2) with U, V being unitary matrices. The matrix product UH(In−A)−1BV can be rewritten as (u1|U2)(In −A)−1B(v1|V2) =( uH1 (In −A)−1Bθ1 uH1 (In −A)−1BV2 U2(In −A)−1Bθ1 UH 2 (In −A)−1BV2 ) = ( σ1u H 1 u1 uH1 (In −A)−1BV2 σ1U H 2 u1 UH 2 (In −A)−1BV2 ) = ( σ1 wH 0 B ) , where uH1 u1 = 1, UH 2 u1 = 0, w = V H 2 ((In −A)−1B)Hu1, and C = UH 2 (In −A)−1BV2. Let w = 0, we get σ2 1 = ∥(In−A)−1B∥22 = ∥UH(In−A)−1BV ∥22 = max x ̸=0 ∥UH(In −A)−1BV x∥22 ∥x∥22 = max x ̸=0 ∥ ( σ1 uH 0 C ) x∥22 ∥x∥22 . Take x −→ w, we have that σ2 1 > (σ2 1 + wHw)2 (σ2 1 + wHw) = σ2 1 + wHw. Thus, this yield that w = 0, and UH(In −A)−1BV = ( σ1 0 0 C ) or (In −A)−1B = U ( σ1 0 0 C ) V H . M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 6 of 17 We see that µs((In −A)−1B) < σ1((In −A)−1B), we have that (In −A)−1B := S−1(In −A)−1B = ( ((In −A)−1B)11 ((In −A)−1B)12 1 ν ((In −A)−1B)21 1 ν ((In −A)−1B)22 ) . Furthermore,( I (In −A)−1B ((In −A)−1B)H I ) > 0 ⇐⇒ I − (In −A)−1BI−1((In −A)−1B)H > 0. From this we follow that λi(I − (In −A)−1B((In −A)−1B)H(ν)) > 0, ∀i or 1− λi((In −A)−1B((In −A)−1B)H(ν) > 0,∀i or λi((In −A)−1B((In −A)−1B)H(ν)) < 1, ∀i. Thus finally, we get σ1((In −A)−1B) < 1 ⇒ ||(In −A)−1B||2 < 1 ⇒ ρ((In −A)−1B)) < 1. Theorem 4 gives the dynamic D-stability of dynamical model yt = (In−A)−1B+Ext. We make use of results on interconnection between structured singular value, and D- stability. Theorem 4. Let the dynamical system be yt = (In−A)−1B+Ext. Then, for dynamic D- stability the matrix (In−A)−1B is D-stable iff (In−A)−1B is stable, and 0 ≤ µ∆(M) < 1, where M := ( iIn + (In −A)−1B )−1 ( iIn − (In −A)−1B ) . Proof. The matrix (In − A)−1B is D-stable iff 0 ≤ µ∆(M) < 1. Assume that matrix M is D-stable, means that, λi ( In − (In −A)−1B + iP ) ̸= 0, ∀ i = 1 : n, and for some P = Diag(Re(pii)) > 0, ∀ i = 1 : n. In order to prove that (In − A)−1B is D-stable matrix iff 0 ≤ µ∆(M) < 1, we let (In −A)−1B is D-stable matrix, that is, λi ( (In −A)−1B + iP ) ̸= 0, ∀ i = 1 : n. Consider a block-diagonal matrix ∆̂ = (iIn − P )(iIn + P )−1, ∆̂ ∈ ∆. This allow us P = (iIn+∆̂)−1(iIn−∆̂) is a positive diagonal matrix if ∆̂ ∈ ∆. Since, λi ( (In −A)−1B + iP ) ̸= 0, ∀ i = 1 : n. Thus, it shows that λi ( (In −A)−1B + i(iIn + ∆̂)−1(iIn − ∆̂) ) ̸= 0, ∀ i = 1 : n, ∀ ∆̂ ∈ ∆. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 7 of 17 As, the rank of matrix ( (In −A)−1B + i(iIn + ∆̂)−1(iIn − ∆̂) ) is exactly equal to the rank of matrix ( (iIn + (In −A)−1B)− (iIn − (In −A)−1B)∆̂ ) , ∀ ∆̂ ∈ ∆. Also,( (In −A)−1B + i(iIn + ∆̂)−1(iIn − ∆̂) ) ∼ ( (iIn + (In −A)−1B)− (iIn − (In −A)−1B)∆̂ ) , ∀ ∆̂ ∈ ∆. Furthermore, λi ( In − (iIn + (In −A)−1B)−1(iIn − (In −A)−1B)∆̂ ) ̸= 0, ∀∆̂ ∈ ∆. This is a necessary condition that 0 ≤ µ∆(M) < 1. Since, our aim is to show that (In −A)−1B is D-stable matrix. This means that we need to show λi ( (In −A)−1B + iP ) ̸= 0, ∀ i = 1 : n. Since, 0 ≤ µ∆(M) < 1, means that λi(iIn − M∆̂) ̸= 0, ∀ ∆̂ ∈ ∆. In turn this implies that λi ( In − (iIn + (In −A)−1B)−1(iIn − (In −A)−1B)∆̂ ) ̸= 0, ∀∆̂ ∈ ∆ which further reduces to the fact that λi ( (In −A)−1B + iP ) ̸= 0, and this shows that (In −A)−1B is a D-stable matrix. Theorem 5. Let the dynamical system be yt = (In−A)−1B+Ext. Then, for dynamic D- stability the matrix (In−A)−1B is D-stable iff Re ( λi(P (In −A)−1B + ((In −A)−1B)TP ) ) > 0, ∀ i = 1 : n, and 0 ≤ µ∆(M) < 1, with M := ( iIn + P (In −A)−1B + ((In −A)−1B)TP )−1 ( iIn − P (In −A)−1B − ((In −A)−1B)TP ) . Proof. We follow the same procedure as given in Theorem 4 to prove Theorem 5. We aim to show that (In−A)−1B isD-stable iffRe ( λi(P (In −A)−1B + ((In −A)−1B)TP ) ) > 0, ∀ i = 1 : n. Let ∆̂ ∈ ∆ be a block-diagonal structure, and defined as ∆̂ := (iIn − P )(iIn + P )−1, a diagonal matrix. As λi(P (In − A)−1B + ((In − A)−1B)TP ) ̸= 0, ∀i = 1 : n. This implies that λi(P (In − A)−1B + ((In − A)−1B)TP + iP ) ̸= 0, ∀i = 1 : n iff λi(P (In−A)−1B+((In−A)−1B)TP + i(iIn+∆̂)−1(iIn−∆̂)) ̸= 0, ∀i = 1 : n. This further reduces to λi ( iIn + P (In −A)−1B + ((In −A)−1B)TP )− (iIn − P (In −A)−1B − ((In −A)−1B)TP )∆̂ ) ̸= 0. This, finally we have that λi ( In − (iIn + P (In −A)−1B + ((In −A)−1B)TP )−1(iIn − P (In −A)−1B − ((In −A)−1B)TP )∆̂ ) ̸= 0. This last inequality is the necessary condition that structured singular values is strictly less than 1, means that, 0 ≤ µ∆(M) < 1. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 8 of 17 3. Necessary and sufficient conditions for D-stability In this section, we present some new results on necessary and sufficient conditions for D-stability of a given matrix in term of its structured singular values. Lemma 1. [20]. A ∈ Rn,n which is continuous-time diagonal stable matrix is a D-stable matrix. Lemma 2. [20]. A ∈ Rn,n which is discrete-time diagonal stable matrix is a D-stable matrix. The following Theorem 6 show that Hurwitz-stable matrix is also a continuous-time D-stable matrix matrix. Theorem 6. [20]. Let A ∈ Rn,n be a Hurwitz-stable matrix. Then A is continuous-time D-stable only if 0 ≤ µ∆ ( (sIn +A)(sIn −A)−1 ) ≤ 1, ∀ s ∈ C+. From above Theorem 6, it is clear that the definition of structured singular value holds true for all the values of parameter s ∈ C+, that is, in closed right-half of complex plane. The following lemma show that structured singular value can be determined at a single value of s ∈ C+ rather than evaluating at entire closed right-half of complex plane. Lemma 3. [20]. Let A ∈ Rn,n be a Hurwitz-stable matrix. Then, A is continuous-time D-stable matrix iff 0 ≤ µ∆ ( (iIn +A)(iIn −A)−1 ) ≤ 1, i = √ −1. Theorem 7 present an interesting relation between a continuous-time D-stable matrix A ∈ Rn,n and structured singular values of a perturbed matrix obtained from A ∈ Rn,n. Theorem 7. Let A ∈ Rn,n such that Re(λi(A)) > 0, ∀ i and is continuous-time D-stable matrix, then 0 ≤ µ∆ ( (αIn +A)−1(αIn −A) ) ≤ 1, α ∈ C+. Proof. Let A = eH be a stable matrix. The matrix H ≥ 0, a positive semi-definite matrix. Let P > 0, a positive definite such that λi(αIn + eHP ) ̸= 0, ∀i, and P = (αIn + ∆̂)−1(αIn − ∆̂) for all ∆̂ ∈ ∆. This formulation allows as to have that λi ( αIn + eH(αIn + ∆̂)−1(αIn − ∆̂) ) ̸= 0 ∀i, ∀∆̂ ∈ ∆. The above expression for λi reduces to λi ( (αIn + eH)−1(αIn − eH)∆̂ ) ̸= 0 ∀i, ∀∆̂ ∈ ∆. In turn this yields λi ( (αIn +A)−1(αIn −A)∆̂ ) ̸= 0 ∀i, ∀∆̂ ∈ ∆. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 9 of 17 Finally, we conclude that 0 ≤ µ∆ ( (αIn +A)−1(αIn −A) ) ≤ 1. The following Theorem 8 gives an interconnection between continuous-time D-stable matrix and structured singular values of a perturbed matrix. Theorem 8. Let A ∈ Rn,n such that Re(λi(A)) > 0, ∀ i and is continuous-time D-stable matrix, then 0 ≤ µ∆ ( (iIn +A)−1(iIn −A) ) < 1, i = √ −1. Proof. We aim to show that Re(λi(A)) > 0,∀i if Re(λi(AH)) = Re(λi(H)),∀i,∀H ≥ 0. If Re(λi(A)) ≥ 0, ∀i and A ∈ Cn×n is n × n-singular matrix, then there exists a unitary matrix U such that U∗AU = ( M11 + iN11 iN12 iN21 · ) , with M11 > 0, and for U∗AU = ( · · · In ) ≥ 0. In turn, this yields Re(λi(AH)) = Re(λi(H)), ∀i. Secondly, we prove that 0 ≤ µ∆ ( (iIn +A)−1(iIn −A) ) < 1. To prove the above result, we take the given matrix A = eH , a stable matrix where H ≥ 0, a positive semi-definite matrix. Let P > 0, a positive definite such that λi(iIn+eHP ) ̸= 0, ∀i where P = (iIn + ∆̂)−1(iIn − ∆̂) for all ∆̂ ∈ ∆. This allows as to have that λi ( iIn + eH(iIn + ∆̂)−1(iIn − ∆̂) ) ̸= 0 ∀i,∀∆̂ ∈ ∆. The above expression takes the form λi ( (iIn + eH)−1(iIn − eH)∆̂ ) ̸= 0 ∀i,∀∆̂ ∈ ∆. Finally, this implies that λi ( (iIn +A)−1(iIn −A)∆̂ ) ̸= 0 ∀i,∀∆̂ ∈ ∆. Thus, 0 ≤ µB ( (iIn +A)−1(iIn −A) ) < 1. Theorem 9. Let A ∈ Cn×n satisfies Re(λi(A)) > 0, ∀ i is a continuous-time diagonal stable matrix and its dual matrix  := (A− I)−1(A+ I) is discrete-time diagonal stable, then σmax(Â1) < 1, with Â1 := D 1 2 ÂD −1 2 . M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 10 of 17 Proof. The matrix A ∈ Cn×n is continuous-time diagonal matrix. This means that for a positive diagonal matrix D, the matrix DA+ATD is negative definite, that is, DA+ATD < 0 ⇐⇒ D(Â+ I)(Â− I)−1 + ((Â+ I)(Â− I)−1)TD < 0 ⇐⇒ D 1 2 (Â+ I)(Â− I)−1D −1 2 + ((Â+ I)(Â− I)−1)TD < 0 ⇐⇒ (Â1 + I)(Â1 − I)−1 + ((Â+ I)(Â− I))−1D < 0 ⇐⇒ Â1 T Â1−I < 0 ⇐⇒ λmax((Â1 T )Â1−I) < 0 ⇐⇒ ρ((Â1 T )Â1) < 1 ⇐⇒ σmax(Â1) < 1. 3.1. Pseudo-spectrum The computation of pseudo-spectrum for a given matrix (say) M is the set containing the all eigenvalues of M . One may raise an important question about the singularity of given matrix M as it does not appear as a small perturbation ϵ which may completely change the answer from a yes to a no. This further implies that either matrix-norm ||M−1|| is large enough or not? For an eigenvalue λ corresponding to given matrix M , a much important question one may ask: Is the matrix ||(λIn − M)−1|| is large or not? this pattern allows to have definitions and results of pseudo-spectrum given as below: Definition 9. For matrix n-dimensional matrix M , and for ϵ > 0, a small perturbation level. The ϵ-pseudospectrum σϵ(M) is the set of eigenvalues λ ∈ C so that ||(λIn −M)−1|| > 1 ϵ . Remark 1. For quantity λ ∈ σ(M), σ(M), denotes the set of eigenvalues of M , ||(λIn − M)−1|| = ∞. The second definition of pseudo-spectrum is given as follows. Definition 10. For an n-dimensional matrix, and for a given ϵ > 0, a small perturbation level. The ϵ-pseudospectrum σϵ(M) is the set of eigenvalues λ ∈ C so that λ ∈ σ(M + E), for some E having ||E|| < ϵ. The third characterization of the computation of pseudo-spectrum for given matrix M is given as bellow. Definition 11. For a given n-dimensional matrix M , and ϵ > 0, a small perturbation level. The ϵ-pseudo spectrum σϵ(M) is the set of eigenvalues λ ∈ C so that ||(λIn −M)v|| < ϵ for some v ∈ Cn,1, ||v|| = 1. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 11 of 17 The following Theorem gives an equivalence of all above definitions of pseudo-spectrum. Theorem 10. Consider that || · || denotes a matrix norm for a given matrix M . Following statement are equivalent: (i) Λϵ(M) = {z ∈ C : ||(zIn −M)−1|| ≥ 1 ϵ }. (ii) Λϵ(M) = {z ∈ C : z ∈ Λ(M + E), ||E|| ≤ ϵ}. (iii) Λϵ(M) = {z ∈ C : ∃ v ∈ Cn,1 s.t ||(M − zIn)v|| ≤ ϵ}. Remark 2. The second statement in the above theorem is true for some matrix E. Fur- thermore, in last statement the column vector v has a unit 2-norm, that is, ||v||2 = 1. 4. Numerical Experimentation In this section, we a present a comparison on the numerical computation of lower bounds of structured singular values. The numerical algorithms under consideration for approximation of lower bounds of structured singular values are: The Matlab function mussv, the power algorithm (PA) [32], Gain Based Algorithm (GBA) [39], Poles mi- gration Algorithm (PMA) [29], Non-linear optimization Algorithm (NLA) [19], and the Low-rank ODE’s based Algorithm (LRA) given by first author [18]. The matrices are taken from various models of economy and finance. Furthermore, we use EigTool [28] for the computation of the pseudo-spectrum of each matrix. Example 1. Consider macroeconomic model of the trade cycle [7] DK = γ1(K̂ −K)with K̂ = β1Y DC = γ2(Ĉ − C)with Ĉ = β2Y + β3 DY = γ3(Ŷ − Y )with Ŷ = C +DK Here, K,C, Y denotes stock of capital, consumption, and output less replacement, respec- tively. Case-I: For γ1 < 0.0625, γ2 = 0.6, γ3 = 4.0, β1 = 2.0, β2 = 0.75, the matrix C has the structure: C = −0.05 0 0.1 0 −0.6 0.45 −0.2 4.0 −3.6  . We present the comparison on numerical approximation of the lower bounds of structured singular values in following Table 1. The numerical approximation of lower bounds of structured singular values mussv PA GBA PMA NLA LRA 5.4369 5.4369 5.4389 5.4391 5.4371 5.4370 Case-II: For γ1 = 0.4, the matrix C has the structure: M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 12 of 17 Figure 1: The graphs of singular values and pseudo-inverse of M in Example-1 (Case-I) Figure 2: The graphs of singular values and pseudo-inverse of M in Example-1 (Case-II) C = −0.4 0 0.8 0 −0.6 0.45 −1.6 4.0 −0.8  . We present the comparison on numerical approximation of the lower bounds of structured singular values in following Table 2. The numerical approximation of lower bounds of structured singular values mussv PA GBA PMA NLA LRA 4.4272 4.4274 4.4285 4.4293 4.4290 4.4272 Example 2. Consider linear dynamical model yt = (In −A)−1B + Ext. For A = [ 0 1 0 0 ] , B = [ 0 0.5 0 0 ] . The matrix (In −A)−1B for n = 2, has the following structure: (I2 −A)−1B = [ 0 0.5 0 0 ] . We present the comparison on numerical approximation of the lower bounds of structured singular values in following Table 3. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 13 of 17 Figure 3: Matlab interface for computing pseudo-spectrum of matrix (I2 −A)−1B. The numerical approximation of lower bounds of structured singular values mussv PA GBA PMA NLA LRA 0.5000 0.5123 0.5341 0.5001 0.5012 0.5000 Example 3. Consider the non-linear model of macro-econometrics presented in [6]. The matrix C = (In −A)−1B and has the following structure for n = 13: −0.08 0 0 0 0 0 0 −0.01 0.01 0 0 0 0 −1.27 0 0 0 0 0 0 −0.01 0.01 0 0 0 0 0 0 −0.24 0 0.24 0 0 0 0 0 0 0 0 0 0 0 −0.09 0.10 0 0 0 0 0 0 0 0 7.63 0 0.48 0 −0.65 0 0.11 0 0 0 0 0.05 0 8.14 0 0.51 0 0 −0.35 0.12 0 0 0 −0.33 0.05 0 0 0 0 0 0 0 −0.29 −0.29 0 0 0 0 0 0 0 0 0 0.02 0 0 −0.13 0.13 0 0 −0.02 0 0 0 0 0 0.11 0 0 0 0 0 0 −0.01 0 0 0 0 0 0.19 0 0 0.19 0 −0.16 0 0 −0.19 −8.60 0 −0.54 0 0.60 0.13 −0.13 0 0 0 0 −0.05 0 1.00 0 0 0 −0 0 0 0 0 0 0 0 0  . We present the comparison on numerical approximation of the lower bounds of structured M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 14 of 17 Figure 4: Matlab interface for computing pseudo-spectrum of matrix (I2 −A)−1B. singular values in following Table 4. The numerical approximation of lower bounds of structured singular values mussv PA GBA PMA NLA LRA 14.2303 14.2309 14.2312 14.2332 14.2398 14.2306 5. Conclusion In this article, we have developed new results on stability, and D-stability of linear economic models. The new results are obtained by using tools from linear algebra, matrix analysis, and system theory. Some novel results are also presented on necessary and suffi- cient conditions on the interconnection between D-stable matrices and structured singular values. The numerical experimentation show the comparison of structured singular values by various numerical techniques, the EigTool is used to present the pseudo-spectrum of matrices across linear dynamic models. The main advantages of the proposed methodol- ogy in the present study: 1. The proposed methodology helps to study and analyze many spectral properties of structured matrices. It contains the properties like eigenvalues, singular values, struc- tured singular values. 2. The proposed methodology based on theoretical results link the bridge between sta- bility, D-stability, and structured singular values for structured matrices corresponding to dynamical systems. 3. The geometrical interpretation gives an advantage to exploit the hidden structures of structured matrices. 4. The proposed methodology has strong theoretical foundations and also numerical ex- perimentation to support the theoretical construction. Acknowledgements This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-23-DR-128). Therefore, the authors thank the University of Jeddah for its technical and financial support. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 15 of 17 References [1] Aneya, Y.P.; Nair, K.P.K. Bi-criteria transportation problem. Manag. Sci. 1979, 25, 73–78. [2] Arrow, K. J. and McManus, M. (1958), A note on dynamic stability, Econometrica, 26, 448-454. [3] A. Bhaya and E. Kaszkurewicz, Linear Algebra & its Applications, vol. 187, pp. 87–104, 1993. [4] A. Berman and D. Hershkowitz, SIAM J. on Algebra & Discrete Methods, vol. 4, pp. 377–382, 1983. [5] Bergstrom, A.R., K.B. Nowman and C.R. Wymer, 1992, Gaussian estimation of a sec- ond order continuous time macroeconometric model of the UK, Economic Modelling 9, 313-351. [6] Bergstrom, A.R. and C.R. Wymer, 1976, A model of disequilibrium neoclassical growth and its application to the United Kingdom, in: A.R. Bergstrom, ed., Sta- tistical inference in continuous time economic models (North-Holland, Amsterdam) 267-327. [7] Bergstrom, Albert R. Continuous time stochastic models and issues of aggregation over time. Handbook of econometrics 2 (1984): 1145-1212. [8] C. R. Johnson, J. of Econ. Theory, vol. 9, pp. 53–62, 1974. [9] Dai, Liyi, ed. Singular control systems. Berlin, Heidelberg: Springer Berlin Heidel- berg, 1989. [10] D. Yue and Q. Han, Delay-Dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and markovian switching, IEEE Trans. Automat. Control, vol. 50, pp. 217-222, Feb. 2005. [11] Donaghy, K.P., 1993, A continuous-time model of the United States economy, in: (3. Gandoifo, ed., Continuous time econometrics; Theory and applications (Chapman & Hall, London) 151-193. [12] Doyle, J. Analysis of feedback systems with structured uncertainties. IEE Proc. D- Control Theory Appl. 1982, 129, 242–250. [13] Enthoven, A. C. and Arrow, K. J. (1956), A theorem on expectations and the stability of equilibrium, Econometrica, 24, 288-293. [14] Fang, Chun-Hsiung, and Fan-Ren Chang. Analysis of stability robustness for gener- alized state-space systems with structured perturbations. Systems & Control Letters 21, no. 2 (1993): 109-114. [15] Fang, Chun-Hsiung, Li Lee, and Fan-Ren Chang. Robust control analysis and design for discrete-time singular systems. Automatica 30, no. 11 (1994): 1741-1750. [16] Fisher, F.M., 1965, Choice of units, column sums, and stability in linear dynamic systems with nonnegative square matrices, Econometrica 33, 445-450. [17] G. P. Barker, A. Berman, and R. J. Plemmons, Linear & Multilinear Algebra, vol. 5, pp. 249–256, 1978. [18] Guglielmi, Nicola, Mutti-Ur Rehman, and Daniel Kressner. A novel iterative method to approximate structured singular values. SIAM Journal on Matrix Analysis and M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 16 of 17 Applications 38.2 (2017), 361-386. [19] Halton, M., Hayes, M., & Iordanov, P. State-space µ analysis for an experimental drive-by-wire vehicle. International Journal of Robust and Non-linear Control, 18(9) (2008) 975–992. [20] Kim, Kwang-Ki K., and Richard D. Braatz. Continuous-and discrete-time D-stability, joint D-stability, and their applications: µ theory and diagonal stability approaches. 2012 IEEE 51st IEEE Conference on Decision and Control (CDC). IEEE, 2012. [21] Kemp, M.C. and Y. Kimura, 1978, Introduction to mathematical economics (Springer-Verlag, New York). [22] Kushel, O.Y., Pavani, R. Novel Versions of D-Stability in Matrices Provide New Insights into ODE Dynamics. Mediterr. J. Math. 20, 225 (2023). [23] Kaczorek, Tadeusz. Analysis and comparison of the stability of discrete-time and continuous-time linear systems. Archives of Control Sciences 26, no. 4 (2016). [24] Lewis, Frank L. A tutorial on the geometric analysis of linear time-invariant implicit systems. Automatica 28, no. 1 (1992): 119-137. [25] Leontief, W. Input-Output Economics. (1986). [26] McKenzie, L., 1960, Matrices with dominant diagonals and economic theory, in: K.J. Arrow, S. Karlin and P. Suppes, eds., Mathematical methods in the social sciences, 1959 (Stanford University Press, Stanford) 47-62. [27] Murata, Y., 1977, Mathematics for stability and optimization of economic systems (Academic Press, New York). [28] Mark Embree and Lloyd N. Trefethen. Pseudospectra Gateway. [29] Magni, J., Döll, C., Chiappa, C., Frappard, B., & Girouart, B. Mixed µ-analysis for flexible systems. Part 1: Theory. In Proceedings of the 14th IFAC world congress, Beijing, China, (1999), 325–360. [30] Nieuwenhuis, Herman J., and Lambert Schoonbeek. Stability and the structure of continuous-time economic models. Economic Modelling 14, no. 3 (1997): 311-340. [31] O’connor, Robert, and Edmund W. Henry. Input-output analysis and its applications. (1975). [32] Packard, Andy, Michael KH Fan, and John Doyle. A power method for the structured singular value. IEEE Conf. on Decision and Control, (1988). [33] Packard, Andrew, and John Doyle. The complex structured singular value. Automat- ica 29.1 (1993): 71-109. [34] Rehman, MU., Rasulov, T.H. & Amir, F. D-Stability, Strong D-Stability and µ- Values. Lobachevskii J Math 45, 1227–1233 (2024). [35] S. Xu, P. Van Dooren, R. Stefan and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Trans. Automat. Control,vol. 47, pp. 1122-1128, July 2002. [36] S. Xu and C. Yang, Stabilization of discrete-time singular systems: a matrix inequal- ities approach, Automatica, vol. 35, pp. 1613-1617, 1999. [37] Sjoo B., 1993, A continuous-time econometric model for Sweden based on monthly data, in: (3. Gandolfo, ed., Continuous time econometrics; theory and applications (Chapman & Hall, London) 195-227. M.U.R et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5657 17 of 17 [38] Safonov, M.G. Stability margins of diagonally perturbed multivariable feedback sys- tems. IEE Proc. D (Control Theory Appl.) 1982, 129, 251–256. [39] Seiler, P., Balas, G., & Packard, A. A gain-based lower bound algorithm for real and mixed µ problems. In Proceedings of the 45th IEEE conference on decision and control, San Diego, California, (2006), 3548–3553. [40] Tong, Yuhao, and Steven W. Su. Sufficient D-Stability Conditions for Non-Square Matrices. arXiv preprint arXiv:2406.15440 (2024). [41] Woods, J.E., 1978, Mathematical economics (Longman, New York). [42] Xu, Shengyuan, and James Lam. Robust stability and stabilization of discrete singular systems: an equivalent characterization. IEEE Transactions on Automatic Control 49, no. 4 (2004): 568-574. [43] X. Mao, Robustness of exponential stability of stochastic differential delay equation, IEEE Trans. Autom. Control, vol. 41, no. 9, pp. 442-447, Mar. 1996. [44] Y. Kimura, J. of Math. Econ., vol. 8, pp. 113–120, 1981. [45] Z. Wang, H. Qiao, and K. J. Burnham, On stabilization of bilinear uncertain time- delay stochastic systems with markovian jumping parameters, IEEE Trans. Automat. Control, vol. 47, pp. 640-646, April 2002.