EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5637 ISSN 1307-5543 – ejpam.com Published by New York Business Global Spectral Properties of Structured Matrices in Transportation Problems Mutti-Ur Rehman1, Behkzod Aminov2, Mohammed N. Alshehri3, Mustafa M. Mohammed4, Arafa O. Mustafa5,∗, Nhla A. Abdalrahman5, Mona Magzoub6, Sakeena E. M. Hamed5, Runda A. A. Bashir4, Awad A. Bakery4,7,∗ 1 Center of Research and Innovation, Asia International University, Yangiobod MFY, G‘ijduvon Street, House 74, Bukhara, Uzbekistan 2 School of Engineering, Central Asian University, Tashkent, 100027, Uzbekistan 3 Department of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia 4 University of Jeddah, Applied college at Khulis, Department of Mathematics, Jeddah, Saudi Arabia 5 University of Jeddah, College of Business at Khulis, Jeddah, Saudi Arabia 6 Mathematics Department, Applied college at Alkamil, University of Jeddah, Saudi Arabia 7 Ain Shams University, Faculty of Science, Department of Mathematics, Cairo, Abbassia, Egypt Abstract. The Hitchcock-Koopmans transportation problem is a well-known and fundamental op- timization problem which focuses on minimization of the objective function which is basically the transportation cost from multiple sources to multiple destinations. In this article, we present some novel results on the spectral properties of structured matrices appearing in Hitchcock-Koopmans transportation problems. The results on the computation of singular values are presented with usage of tools from linear algebra and matrix analysis. The new results are derived on intercon- nection between structured singular values of pseudo-inverse and D-stable matrices of Hitchcock- Koopmans transportation models. The numerical experimentation shows the behavior of singular values. The Matlab EigTool is used for the computation of pseudo-spectrum of pseudo-inverse matrix corresponding to the transportation model. 2020 Mathematics Subject Classifications: 15A18, 15A16 15A23 Key Words and Phrases: Hitchcock-Koompan model, singular values, structured singular val- ues, D-stable matrices, pseudo-spectrum ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5637 Email addresses: muttiur.abbasi@oxu.uz (Mutti-Ur Rehman), b.aminov@centralasian.uz (Behkzod Aminov), mnalshhri@nu.edu.sa (Mohammed N. Alshehri), mustasta@yahoo.com and mmibrahim@uj.edu.sa (Mustafa M. Mohammed), arafaomustafa2020@yahoo.com and 04220355@uj.edu.sa (Arafa O. Mustafa), 04220332@uj.edu.sa (Nhla A. Abdalrahman), MMAHMED@uj.edu.sa (Mona Magzoub), 04220347@uj.edu.sa (Sakeena E. M. Hamed), rabasher@uj.edu.sa (Runda A. A. Bashir), aabhassan@uj.edu.sa (Awad A. Bakery) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 2 of 20 1. Introduction The transportation theory is a name given to study of optimal transportation and the allocation of resources. In optimization, the transportation problem was earlier formalized by a French mathematician Gaspard Monge, and then A.N. Talstoi first studied transport problem mathematically in 1920 and published an article with titled: Methods of finding minimal kilometrage in the Cargo-Transportation in space. The transportation problem for m numbers of sources x1, x2, · · ·, xm for a given com- modity having a(xi) number of units of supply at xi points and n number of sinks y1, y2, · · ·, yn for commodity. The demand at yj is considered as b(yj). Let xi and yi, a(xi, yi) represents the unit cost of shipment. Then, the question of finding flow satisfying the demand from supplies, and to minimize the cost was studied in [15, 16, 18]. Transportation problems can be considered as an optimization problems, particularly the linear optimization. In linear optimization the aim could be in finding effective tech- niques in order to distribute goods from many suppliers to many final destinations. The mathematical concepts dealing with transportation problems involves the analysis of struc- tured and, unstructured matrices representing supply, demand, and transportation costs. The notable methods likewise Northwest Corner Rule, Least Cost technique, and Vogel’s Approximation Method are developed to find initial feasible solutions corresponding to the optimization problems. On the other hand, the optimization methods for instance Modified Distribution (MODI) Method or stepping-stone method are used to analyze and refine obtained solutions for the purpose of achieving optimality conditions. The transportation problems have an extensive amount of applications in many diverse range of research directions [2, 10]. In [4], the service network design problems were studied in order to measure the minimum cost with given constraints. To study the evolution of coalition over a given time with trust-related issues, an agent-based model was developed in [45]. The traditional transportation models deals with the problems like transportation costs, delivery routes, production places, and the reduction of carbon emissions [42, 43, 48, 54]. The transportation algorithms [17, 34, 51] were developed to solve practical nature of the problems. A new method was developed in [31] to solve the transport problems on northwest corner method in order to reduce the number of steps to determine the number of iteration given in [29]. A new technique for solving balanced transport problem based on geometric average for transport costs was developed in [8]. A number of transportation problems can be solved by using numeric techniques im- plemented in Matlab software. The northwest corner method was used to analyze the transport of chemical substance of a pharmaceutical company [37]. The Vogel’s approxi- mation and modified distribution method were used to deal with large scale cost matrices [27]. The Vogel’s Approximation Method primarily determine the number of penalties ap- pearing in each row and column of the matrix by computing the difference among minimum and second minimum cost. Further, this method also allocate to the cell having minimum cost across each of the row or column with maximum penalty. Once compared with M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 3 of 20 Northwest Corner Rule, this technique generates an initial solution which is very much near to optimality condition. On the other hand, the Modified Distribution Method is being mainly used for the analysis of the optimization to an initial feasible solution which is first obtained by Vogel’s approximation, and Northwest Corner techniques. The main advantage of Modified Distribution Method compared to Northwest Corner Method is the insurance that an optimal solution has reached up to a desired level. The Vogel’s Approximation Method (VAM), Modified Distribution Method (MODI), and Northwest Corner Method (NWC) are mainly used to deal with analysis and the so- lution of transportation problems appearing in the operations research, specifically for the optimization of the logistics. The NWC method is being used as an initial approach for the location of resources when the consideration of cost function are much smaller than critical values. VAM mainly does focus on penalties. It helps to finds applications once the minimization of transportation cost function is in a crucial stage. MODI is being used to ensure the efficiency of cost solution. It does makes its vital role to optimize the trans- portation networks at very large scale, to balance both supply and demands. Furthermore, it ensure the cost efficiencies across the industries, for instance, the manufacturing, retail industry. A new mathematical technique [11] based on simplex algorithm was developed to solve transport problem. A bi-criterion transportation problem was solved by Aneya and Nair [1]. In [28, 52], the optimization of passenger transport problems and passenger control flow problems were studied and analyzed. An extensive amount of literature has been written to deal with time-minimization transport problems, we refer interested readers to see [5, 24, 40, 46, 47] and the references therein. The structured singular values are non-negative numbers obtained by computing the singular values of perturbation matrix ∆̂ from the set of block-diagonal matrices ∆. The structured singular values were first introduced by Doyle [12] to study and investigate both stability and instability of feedback systems. The computation of structured singular is a NP-hard problem [7]. The global search numerical and analytical techniques were developed [19, 32] to deal with lower dimensional mathematical problems. The concept of D-stability for the very first time was introduced by Arrow and Mc- Manus [3], and then Enthoven and Arrow [14] in their classical papers. The main aim in these classical papers was to study and investigate the dynamic models from the eco- nomics. The theory of D-stability plays an important role in a vast amount of application areas in economic analysis [20, 22, 25, 26, 33, 38, 53]. The main objective of this paper is two-fold: First to study and analyze the spectral properties of structured matrices appearing across the Hitchcock-Koopmans transporta- tion models. The computation of various tools like eigenvalues, singular values, right and left singular vectors, structured singular values, and pseudo-inverse describes various spec- tral properties of structured matrices under consideration. On the other hand, we present new results on the interconnection between structured singular values, and D-stability of structured matrices across Hitchcock-Koopmans transportation problem. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 4 of 20 1.1. Overview of the article. In section 2, we provide new results on singular values of a matrix corresponding to Hitchcock-Koopmans transportation problem. We have employed some tools from linear algebra and matrix analysis to develop these results. Furthermore, we also provide numerical experimentation on the computation of singular values and pseudo-inverse of transportation matrices. In section 3 of this article, we provide some new results on the interconnection between structured singular values of (M∗M)−1M∗ and M∗(MM∗)−1. The numerical experimentation on comparison of bounds of structured singular values are presented in section 4, and finally we have presented conclusion in section 5. 2. Singular values for Hitchcock-Koopmans transportation problem The Hitchcock-Koopmans transportation problem was first developed by Hitchcock in 1941, and then investigated and analyzed by Koopmans in 1947. This problem was applied to simplex algorithm by Dantzig in 1951. For m numbers of origins and n numebrs of destinations, the Hitchcock-Koopmans transportation is to study and solve following optimization problem: min{cTx : Mx = g, 1ma = 1nb, x ≥ 0}, where min is taken over x, also aT = [a1, a2, · · ·, am], bT = [b1, b2, · · ·, bm], xT = [x11, x12, · · ·, xmn], cT = [c11, c12, · · ·, cmn], and gT = [aT : bT ]. The coefficient matrix M is given as M = ( 1n ⊗ Im In ⊗ 1n ) , where Im is m-dimensional identity matrix , 1m is 1 ×m vector of all 1′s, and ⊗ denotes kronecker product. In [6], a relation between eigenvectors of M+M and MMT was developed. The Moore- Penrose inverse M+ for M was obtained in [9] as M+ = 1 mn ( m1Tn ⊗ ((m + n)Im − Jm)n((m + n)In) − Jn ⊗ 1Tm ) , with Jm as an m×m matrix of all 1′s, means that, Jm = 1Tm1m. Furthermore, M+M = I − 1 mn (nIn − Jn) ⊗ (mIm − Jm) , with I −M+M, a symmetric and idempotent matrix. The relation between eigenvectors of Jn, Jm, and I −M+M was developed in the Theorem 2 [6]. The following theorem gives Moore-Penrose inverse of a matrix M . M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 5 of 20 Theorem 1. Let M ∈ Rn,n. Then, then M+ is the Moore-Penrose inverse satisfying the matrix equation M = MM+M. Theorem 2. Let vj be all eigenvectors of Jm, and uk, the eigenvectors corresponding to Jn. Then eigenvectors corresponding to simple eigenvalues 1 of I −M+M are v̂i = uj ⊗ xk, where j = 1 : n− 1; k = 1 : m− 1; i = m + n, · · ·, (m− 1)(n− 1). Definition 1. The singular value of a square or rectangular matrix M are the positive square root of eigenvalue of MTM, where T denotes the transpose of M . Theorem 3. Let M ∈ Rn,n. Then there exists unitary matrices U, V , and a diagonal matrix Σ such that M can be decomposed as M = UΣV T . Further, the non-zero singular values of M are contained along the main diagonal of Σ. Corollary 1 [6]. Let {a1, a2, · · ·, am+n−1} and {b1, b2, · · ·, bm+n−1} are the sets of eigenvectors corresponding to eigenvalues of MMT and MTM , and consider that σ1, σ2, · · ·, σm+n−1 are singular values of M . Then, MMTai = σ2 i ai, i = 1 : m + n− 1, MTMbi = σ2 i bi, i = 1 : m + n− 1. Corollary 2 [6]. The eigenvectors ai corresponding to the eigenvalue 0 and m + n are Jmxi = mxi, Jnyi = nyi. The eigenvectors corresponding to eigenvalues m and n are given by Jmxi = 0⃗, Jnyi = 0⃗ with xi are m× 1 vectors and yi are n× 1 vectors. Let M+ = (M∗M)−1M∗ be the Moore-Penrose inverse matrix of the matrix M , ∗ denotes the complex conjugate transpose. Let M∗(MM∗)−1 be the right inverse of M . The following Theorem 4 gives the orthogonal nature of the leading singular values. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 6 of 20 Theorem 4. Let (M∗M)−1M∗ and M∗(MM∗)−1 be n-dimensional matrices. Let {σi}, ∀i = 1 : n be sequence of singular values with {vi}, and {v̂i}, ∀i = 1 : n as the left hand side and right hand side singular vectors such that ||vi||2 = 1 = ||v̂i||2, and {ui}, and {ûi}, ∀i = 1 : n as the left hand side and right hand side singular vectors for {σ̂i} such that ||ui||2 = 1 = ||ûi||2. The leading singular vectors v1 and u1 are orthogonal to σ1 and σ̂1. Then any non-zero vector ṽ = {ṽ1, ṽ2, · · ·, ṽn} is an orthogonal to vector en = 1√ n (1, 1, · · ·, 1)T and is not a singular vector σ1 and σ̂1. Proof. For the prove, we consider singular value problem of form Mṽ = σṽ, where σ is singular value of M corresponding to singular vector ṽ. The vector ṽ doesn’t act as singular vector corresponding to singular values σ1 and σ̂1. The matrix-vector product Mṽ, can be rewritten as Mṽ =  m11ṽ1 + m12ṽ2 + · · ·m1nṽn · · · · · · · · · mn1ṽ1 + mn2ṽ2 + · · ·mnnṽn  . Take ∑ on the components of Mṽ, we have ∑ (Mṽ) = ∑  m11ṽ1 + m12ṽ2 + · · ·m1nṽn · · · · · · · · · mn1ṽ1 + mn2ṽ2 + · · ·mnnṽn  = v1 ∑ (mi1)+···+vn ∑ (min) = v1(σ1)+···+vn(σ1). Thus, ∑ (Mv) = σ1 ∑ (v). Taking ∑ of right hand side of Mv = σv, we get∑ (σv) = σ ∑ (v). We conclude from last two equations that (σ − σ1) = ∑ (v) = 0. This implies that σ ̸= σ1, and ∑ (v) = 0. This proves that v = (v1, v2, · · ·, vn) is orthogonal to vector en = 1√ n (1, 1, · · ·, 1)T . The vector en is not a singular vector corresponding to singular values σ1 and σ̂1. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 7 of 20 Theorem 5. Let (M∗M)−1M∗ and M∗(MM∗)−1 be n-dimensional matrices. Let σi and σ̂i are leading singular values corresponding to singular vectors vi and v̂i, respectively. Let M̂ = ( (M∗M)−1M∗ + αIn 2αv̂2v T 1 2αv1v̂ T 2 M∗(MM∗)−1 + αIn ) , with In is an identity matrix, α is any scalar. The singular values of M̂ does not have leading singular values of matrices (M∗M)−1M∗ and M∗(MM∗)−1. The leading singular values σ1 and σ̂1 are along the main diagonal of M̂1 with M̂1 = ( 3α + σ1 0 0 α− σ̂1 ) . Proof. Consider the singular value problem of the form M̂vi = (σ1 + α)vi, ∀i = 1 : n. Let β1, σ2+α, σ3+α, ···, σn+α, and β̂2, σ̂2+α, σ̂3+α, ···, σ̂n+α with β, β̂ ∈ {3α+σ1, α−σ̂1} are the singular values of matrix ( 3α + σ1 0 0 α− σ̂1 ) . For the singular values σi, ∀i = 2 : n, the singular vectors can be written as [vi 0]T , ∀i = 2 : m. For the singular values σ̂i, ∀i = 2 : n, the singular vectors can be written as [0 v̂i] T , ∀i = 2 : n. This ensures that singular vectors corresponding to M̂ can be ex- pressed as [βivi β̂iv̂i] T . Thus, [βi β̂i], ∀i = 2 : n are the singular vectors corresponding to singular values σ1 + 3α and α− σ̂. The following Theorem shows that singular values of a matrix does not depend con- tinuously on the entries of that matrix. Theorem 6. Let M1 = (M∗M)−1M∗ and M2 = M∗(MM∗)−1 be n-dimensional matrices. Let limk→∞(Mk) = M, and let q = minm, n. Consider that σ1(M) ≥ σ2(M) ≥ · · · ≥ σq(M), and σ1(Mk) ≥ · · · ≥ σq(Mk) be non-increasing ordered singular values of M and Mk, respectively for k = 1, 2, · · ·. Then the limk→∞ σi(Mk) = σi(M) for all i = 1 : q. Proof. Let k1 < k2 < · · ·, be the sequence of positive integers, and let ϵ > 0, we have that max|σi(Mkj) − σi(M)| > ϵ, where max is taken over all i = 1 : q. Consider the singular value decomposition of Mkj as Mkj = UkjΣkjV ∗ kj , where Ukj and Vkj are the unitary matrices and Σkj is a diagonal matrix with structure Σkj = [σ1(Mkj) · · · σq(Mkj)] T . M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 8 of 20 (a) Singular values and pseudo singular values (b) Surface plot of pseudo-spectrum Figure 1: The graphs of singular values and pseudo-inverse of M in Example-1 Then, lim r→∞ Σkjr = lim r→∞ U∗ kjrMkjrVkjr = ( lim r→∞ U∗ kjr )( lim r→∞ Mkjr )( lim r→∞ Vkjr ) = U∗MV. The matrix U∗MV is a non-negative diagonal matrix. The uniqueness of the singular values of M implies that Diag Σ = [σ1(M), σ2(M), · · ·, σq(M)]T , a contradiction with inequality max|σi(Mkj) − σi(M)| > ϵ. This proves required result. Next, we give numerical examples on the computation of singular values and pseudo inverse of transportation matrices. Example 1. Consider 7× 5 transportation matrix (staircase matrix) taken from [16]. M =  1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1  . The computation of singular values and the graphs of the pseudo-inverse are shown in Figure 1. Example 2. Consider 7 × 5 transportation matrix (distribution matrix) taken from [16]. M =  4 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1.4 0.6  . M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 9 of 20 (a) Singular values and pseudo singular values (b) Surface plot of pseudo-spectrum Figure 2: The graphs of singular values and pseudo-inverse of M in Example-2 (a) Singular values and pseudo singular values (b) Surface plot of pseudo-spectrum Figure 3: The graphs of singular values and pseudo-inverse of M in Example-3 The computation of singular values and the graphs of the pseudo-inverse are shown in Figure 2. Example 3. Consider 7 × 5 transportation matrix (distribution matrix) taken from [16]. M =  1.6 0.8 1.6 0 0 0 0 2.4 1.2 2.4 0 0 0 0 0 0 0.6 1.8 0.6 0 0 0 0 0.4 1.2 0.2 0.2 0 0 0 0 0 1.2 1.2 0.6  . The computation of singular values and the graphs of the pseudo-inverse are shown in Figure 3. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 10 of 20 3. Structured singular values for Hitchcock-Koopmans transportation problem In this section, we present new results on the computation of structured singular values for matrices appearing in Hitchcock-Koopmans transportation problems. We make use of mathematical tools from linear algebra, system theory and matrix theory to provide and analyzed the results on structured singular values. Definition 2. The n-dimensional matrix M is stable if all the real parts of the eigenvalues are strictly positive, that is, Re(λi(M)) > 0. Definition 3. The n-dimensional matrix M is D-stable if all the real parts of the eigen- values of MD are strictly positive, that is, Re(λi(MD)) > 0, where D = diag(dii) > 0, for all i = 1 : n. The structured singular value of a given matrix M with respect to set of block-diagonal matrices ∆, where ∆ := {diag (δ1Ir1 , δ2Ir2 , · · ·, δSIrS ; ∆1,∆2, · · ·,∆F ) : δi ∈ R(C), ∆j ∈ Kmj ,mj , i = 1 : S; J = 1 : F}, with K = R(C), is the computation of largest singular value of ∆̂ ∈ ∆. The structured singular value is denoted by µ, and for a given matrix M and ∆, it is defined as (see [12]): µ∆(M) := ( min{||∆̂||2 : det(In −M∆̂) = 0, ∀ ∆̂ ∈ ∆} )−1 , where min is taken over ∆̂ ∈ ∆, and µ∆(M) = 0 if det(In −M∆̂) ̸= 0, ∀ ∆̂ ∈ ∆. Remark 1. The block-diagonal structure ∆ can be associated with multi-index of the positive integers. Remark 2. The full blocks in ∆ can be taken as either pure real blocks, pure complex blocks or a mixture of both. These blocks can be chosen as rank-1 matrices. Remark 3. For any α ∈ C, µ∆(αM) = |α|µ∆(M). Corollary 1. [12] The structured singular value µ∆(M) is equal to the computation of spectral radius ρ of V TMW, that is, µ∆(M) = ρ(V TMV ), with V,W having the block-diagonal structure. The µ∆(M) can be considered as the computation of spectral radius ρ of M∆̂, ∆̂ ∈ ∆. Lemma 1. [35] For M, and a block-diagonal structure ∆, µ∆(M) = maxρ(M∆̂), where max is taken over ∆̂ ∈ ∆. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 11 of 20 Theorem 7 is a well-known result on D-stability of a given matrix while computing the strictly positive real part of the spectrum of the product of given matrix with a positive diagonal matrix. Theorem 7. [14] If M has all negative diagonal elements, and no negative off diagonal elements, for D = diag(dii), the matrices M , and DM are stable, then D = diag(dii) > 0, ∀i. The following theorem 8 is the characterization of D-stability and bridge a link between µ-values and D-stable matrices. Theorem 8. Consider M be a n-dimensional matrix. Then M is D-stable matrix if and only if M is stable and 0 ≤ µ∆(iIn + M)−1(iIn −M) < 1. Theorem 9. Let (MM∗)−1M∗ be n−dimensional complex valued matrix. Then 0 ≤ µ∆((In + (MM∗)−1M∗)−1(In − (MM∗)−1M∗)) < 1. Proof. To prove this result, we use the concept of D−stability of a matrix. That is, the given (MM∗)−1M is D−stable, means that, λk(In + (MM∗)−1Mp) ̸= 0,∀k = 1 : n, with P = diag(p11, p22, . . . , pnn) > 0. Let P = (In + R)−1(In −R),∀R ∈ ∆, then λk(In + (M∗M)−1M∗(In + R)−1(In −R)) ̸= 0, ∀k = 1 : n,∀R ∈ ∆. This further implies that λk((In + (M∗M)−1M∗) + (In − (M∗M)−1M∗R)) ̸= 0, ∀k = 1 : n,∀R ∈ ∆. Since, λk(In+(M∗M)−1M∗) ̸= 0 ∼ λk((In+(M∗M)−1M∗)+(In−(M∗M)−1M∗R)) ̸= 0,∀k = 1 : n, ∀R ∈ ∆ Finally, this yields that 0 ≤ µ∆((In + (MM∗)−1M∗)−1(In − (MM∗)−1M∗)) < 1, which is the required prove. Theorem 10. Let (M∗M)−1M∗ be n−dimensional complex valued matrix. Then 0 ≤ µ∆(A) < 1 if (M∗M)−1M∗ is D−stable, with A = (iIn + P (M∗M)−1M∗ + M((M∗M)−1)∗)(iIn − P (M∗M)−1M∗ −M((M∗M)−1)∗P ) with P = diag(p11, p22, . . . , pnn), pii > 0, ∀i = 1 : n, i = √ −1. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 12 of 20 Proof. The given matrix (M∗M)−1M∗ is D−stable if Re(λk(P (M∗M)−1M∗+M(M∗M−1)∗)) > 0,∀k = 1 : n, see [13]. To prove that 0 ≤ µ∆(A) < 1, we consider a block-diagonal struc- ture ∆̂ = (iIn − P )(iIn + P )−1 with ∆̂ ∈ ∆. For all P , the diagonal positive definite matrices, we have λk(P (M∗M)−1M∗ + M((M∗M)−1)∗P + i(iIn + ∆̂)−1(iIn − ∆̂)) ̸= 0∀k = 1 : n. In turn, the expression for λk,∀k = 1 : n takes the form λk((iIn+P (M∗M)−1M∗+M((M∗M)−1)∗P )−(iIn−P (M∗M)−1M∗−M((M∗M)−1)∗P )∆̂) ̸= 0,∀∆̂ ∈ ∆. Thus, λk(In−(iIn+P (M∗M)−1M∗+M((M∗M)−1)∗P ))(iIn−P (M∗M)−1M∗−M((M∗M)−1)∗P )∆̂) ̸= 0, ∀∆̂ ∈ ∆. The last expression for λk, ∀k = 1 : n implies that 0 ≤ µ∆(A) < 1 Theorem 11. Let (M∗M)−1M∗ be n−dimensional complex valued matrix. Then, 0 ≤ µ∆(A) < 1 if x∗(M((M∗M)−1)∗P 2 + P 2(M∗M)−1M∗)x > 0 for x ∈ Cn,1, and for all P = diag(p11, p22, . . . , pnn) > 0, with A = (iIn + (M∗M)−1M∗)−1(iIn − (M∗M)−1M∗). Proof. The structured singular value is the computation of αmax ≥ 0 such that for each P , the matrix inequality ∥P (M∗M)−1M∗x∥) ∥Px∥ ≥ αmax. For given A, 0 ≤ µ∆(A) < 1 if ∥PAx∥ < ∥Px∥ for every x ∈ Cn,1 and positive diagonal matrix P . The above inequality holds true for iIn + (M∗M)−1M∗, means that, ∥PA(iIn + (M∗M)−1M∗)x∥ < ∥P (iIn + (M∗M)−1M∗)x∥. Also, ∥PA(iIn + (M∗M)−1M∗)x∥2 < ∥P (iIn + (M∗M)−1M∗)x∥2. This further implies that x∗((iIn + (M∗M)−1M∗)∗A∗P ∗PA(iIn + (M∗M)−1M∗))x < x∗((iIn + (M∗M)−1M∗)∗P ∗P (iIn + (M∗M)−1M∗))x. This inequality further reduces to x∗((iIn+(M∗M)−1M∗)∗A∗P 2A(iIn+(M∗M)−1M∗))x < x∗(iIn+(M∗M)−1M∗P 2(iIn+(M∗M)−1M∗))x. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 13 of 20 Since, A = (iIn + (M∗M)−1M∗)−1(iIn − (M∗M)−1M∗). Thus, above inequality rewritten as x∗((iIn+(M∗M)−1M∗)∗(iIn−(M∗M)−1M∗)∗(iIn+(M∗M)−1M∗)−1P 2(iIn+(M∗M)−1M∗)−1 (iIn−(M∗M)−1M∗)(iIn+(M∗M)−1M∗)x−x∗((iIn+(M∗M)−1M∗)∗P 2(iIn+(M∗M)−1M∗))x < 0. or x∗(iIn−(M∗M)−1M∗)∗P 2(iIn−(M∗M)−1M∗)x−x∗(iIn+(M∗M)−1M∗)∗P 2(iIn+(M∗M)−1M∗)x < 0. In turn, we have that x∗((iIn−(M∗M)−1M∗)∗P 2(iIn−(M∗M)−1M∗)−(iIn+(M∗M)−1M∗)∗P 2(iIn+(M∗M)−1M∗))κ < 0. Thus, x∗((iIn−(M∗M)−1M∗)∗(iP 2−P 2(M∗M)−1M∗)−(iIn+(M∗M)−1M∗)∗(iP 2+P 2(M∗M)−1M∗))x < 0. Also, x∗(−2iP 2(M∗M)−1M∗ − 2iM(M∗M)−1P 2)x < 0 or x∗(−2i(M(M∗M)−1)∗P 2 + P 2(M∗M)−1M∗)x < 0. Finally, x∗(M(M∗M)−1)∗P 2 + P 2(M∗M)−1M∗)x > 0, which completes required proof. Pseudo-spectrum: The pseudo-spectrum of a matrix M is the set of which contains the spectrum, that is, all the eigenvalues of M . The important question one can raise is about the singularity of M 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 ||M−1|| is large enough or not? For λ, an eigenvalue of M , a much better question is to ask: Does ||(λIn − M)−1|| is large or not? such a pattern allows following definitions and results [49] of pseudo- spectrum. Definition 4. Let M be a given n-dimensional matrix, ϵ > 0, a small perturbation. The ϵ-pseudospectrum σϵ(M) is the set of eigenvalues λ ∈ C such that ||(λIn −M)−1|| > 1 ϵ . Remark 4. For λ ∈ σ(M), σ(M) being as the set of eigenvalues of M , ||(λIn − M)−1|| = ∞. The second definition of pseudo-spectrum is given as follows. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 14 of 20 Definition 5. Let M be a given n-dimensional matrix, ϵ > 0, a small perturbation. The ϵ-pseudospectrum σϵ(M) is the set of eigenvalues λ ∈ C such that λ ∈ σ(M + E), for some E with ||E|| < ϵ. The third characterization of pseudo-spectrum is given as bellow. Definition 6. Let M be a given n-dimensional matrix, ϵ > 0, a small perturbation. The ϵ-pseudo spectrum σϵ(M) is the set of eigenvalues λ ∈ C such that ||(λIn −M)v|| < ϵ for some v ∈ Cn,1, ||v|| = 1. The following Theorem gives an equivalence of all above definitions of pseudo-spectrum. Theorem 12. [13] Let || · || denotes a matrix norm for a matrix M which is induced by a vector norm. Then following 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 1. In above Theorem 12, the second statement is true for some matrix E. Also, in the last statement the column vector v is such that ||v||2 = 1. 4. Numerical Experimentation We present a comparison on the numerical approximation of structured singular values. The numerical algorithms are: The Matlab function mussv, the power algorithm (PA) [36], Gain Based Algorithm (GBA) [44], Poles migration Algorithm (PMA) [30], Non-linear optimization Algorithm (NLA) [23], and the Low-rank ODE’s based Algorithm (LRA) given by first author [21]. We consider structured matrices appearing across transportation models. The Matlab EigTool [13] is being used for graphical interpretation of the pseudo- spectrum. In each example, we have shown different graphics. The graphical representations in a 2-dimensional space presents the spectrum and pseudo-spectrum of structured matrices. The black dots around the spectrum and pseudo-spectrum denotes the field of values enclosing spectrum and pseudo-spectrum. The visualization of the pseudo-spectrum in the complex plane are level sets of the resolvent norm ||(M − zIn)−1|| which indicates various ϵ-values. These visualizations of the level sets helps to analyze the stability analysis and robustness in various applications like control systems and fluid dynamics. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 15 of 20 (a) Field of values enclosing spectrum and pseudo-spectrum (b) Resolvent norm Figure 4: pseudo-spectrum of A in Example-1 Example 1. Consider 4-dimensional matrix for Traveling Salesman Problem [39]. A =  −1 1 0 0 1 1 0 0 0 1 1 1 3 0 2 1  . 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 3.9984 3.9992 3.9987 3.9990 3.9988 3.9985 Example 2. Consider 6-dimensional symmetric circulant matrix for Travelling Sales- man Problem [50]. A =  0 4 1 6 1 4 4 0 4 1 6 1 1 4 0 4 1 6 6 1 4 0 4 1 1 6 1 4 0 4 4 1 6 1 4 0  . 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 3.9984 3.9992 3.9987 3.9990 3.9988 3.9985 M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 16 of 20 (a) Field of values enclosing spectrum and pseudo-spectrum (b) Resolvent norm Figure 5: pseudo-spectrum of A in Example-2 Example 3. Consider 4-dimensional matrix taken from [41]. A =  3 5 1 −2 0 −1 5 10 3 5 1 9 −2 1 6 6  . We present the comparison on numerical approximation of the lower bounds of structured singular values in following Table-3. The numerical approximation of lower bounds of structured singular values mussv PA GBA PMA NLA LRA 16.4123 16.4176 16.4198 16.4183 16.4142 16.4130 5. Conclusion In this article we have presented new results on the spectral properties (singular val- ues) of structured matrices appearing across Hitchcock-Koopmans transportation mod- els. The interconnection between D-stable matrices and structured singular values of pseudo-inverse are analyzed. The numerical experimentation carried with Matlab shows the dynamics of singular values, structured singular values, and pseudo-spectrum. The advantages of the proposed methodology are listed as: 1. The proposed methodology helps to study the spectral properties via the computation of eigenvalues, eigenvectors, singular values, right and left handed singular vectors, and structured singular values. 2. The proposed methodology has an advantage in the sense that it allow us to establish new interconnections between structured singular values and D-stability analysis of struc- tured matrices appearing across the Hitchcock-Koopmans transportation model. M.U. Rahman et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5637 17 of 20 (a) Field of values enclosing spectrum and pseudo-spectrum (b) Resolvent norm Figure 6: pseudo-spectrum of A in Example-3 3. The computation and graphical representation of spectrum and pseudo-spectrum gives an advantage to exploit the hidden structures and properties of structured matrices. 4. The proposed methodology is well-established in the sense that it has strong theoretical foundations and also numerical experimentation 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. References [1] Aneya, Y.P.; Nair, K.P.K. Bi-criteria transportation problem. Manag. Sci. 1979, 25, 73–78. [2] Angelelli, E.; Morandi, V.; Savelsbergh, M.; Speranza, M.G. System optimal routing of traffic flows with user constraints using linear programming. Eur. J. Oper. 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