EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6417 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Characterization of Diagonal Solutions for a Class of Linear Matrix Inequality Ali Algefary1,∗, Tulin Alhumaidan2 1 Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia 2 Department of Statistics and Operation Research, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia Abstract. This paper investigates the existence of positive diagonal solutions for a class of linear matrix inequalities (LMIs) involving a triple of real n × n matrices (A1, A2, A3). We provide equivalent conditions linking the negative definiteness of a structured block matrix to properties of positive semidefinite test matrices and P -matrices under Hadamard transformations. Our results extend classical stability results to multi-matrix settings with applications to control theory and network dynamics. 2020 Mathematics Subject Classifications: 15A45, 15B48, 93D05, 34K25, 34K05 Key Words and Phrases: Diagonal matrices, P -matrix, positive definite matrix, matrix in- equality, matrix stability 1. Introduction Linear matrix inequalities (LMIs) play a fundamental role in various fields of applied mathematics, including control theory [1–3], optimization [4], and stability analysis of dynamical systems [5]. These inequalities often arise in the study of matrix stability properties, such as Lyapunov stability, and are instrumental in characterizing the behavior of complex systems. A particularly interesting and challenging problem within this domain is the identification of diagonal solutions—specifically, positive diagonal matrices—that satisfy certain LMIs. Such solutions are not only mathematically intriguing but also have practical implications in areas such as network control [6, 7] and evolutionary dynamics [8]. Lyapunov stability is a cornerstone of dynamical systems theory, providing a frame- work to assess the stability of equilibrium points without explicitly solving differential equations [5, 9]. For a linear system ẋ = Ax, where A ∈ Rn×n, the system is said to be Lyapunov stable if there exists a symmetric positive definite matrix P ≻ 0 such that the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6417 Email addresses: a.algefary@qu.edu.sa (A. Algefary), 432206678@qu.edu.sa (T. Alhumaidan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 2 of 10 Lyapunov inequality ATP+PA ≺ 0 holds, ensuring that the quadratic form V (x) = xTPx serves as a Lyapunov function whose time derivative is negative definite along system tra- jectories [1]. This condition guarantees asymptotic stability, meaning solutions converge to the origin as time progresses. A related and more specific concept, Lyapunov diagonal stability, applies when such a P can be chosen as a positive diagonal matrix [10]. This property is particularly significant in the context of linear matrix inequalities (LMIs), as it imposes a structured constraint on the solution space, linking stability to the existence of diagonal matrices that satisfy inequalities like those studied in this work. For a matrix A to be Lyapunov diagonally stable, −A must often exhibit properties such as being a P -matrix [11, 12], a connection that underpins our analysis of diagonal solutions for the triple (A1, A2, A3). The importance of Lyapunov diagonal stability extends beyond theoretical elegance; it arises naturally in applications where system matrices possess inherent structural proper- ties, such as in networked control systems [6, 13] or compartmental models in biology [14]. By restricting P to be diagonal, we enforce a decoupling of variables that simplifies sta- bility analysis and computation, a feature exploited in our characterizations of positive diagonal solutions [15]. This paper builds on these concepts to explore conditions under which a structured block matrix, involving multiple matrices A1, A2, A3, admits such di- agonal solutions, thereby generalizing classical stability results to multi-matrix settings with practical implications in control and dynamics. In this paper, we focus on a specific class of LMIs involving a triple of real n × n matrices (A1, A2, A3). Our objective is to characterize the conditions under which this triple admits a positive diagonal solution, defined as a set of positive diagonal matrices P1, P2, P3 ∈ Rn×n such that the block matrix B = AT 1 P1 + P1A1 + P2 + P3 P1A2 P1A3 AT 2 P1 −P2 0 AT 3 P1 0 −P3  is negative definite. This formulation generalizes classical stability problems and intro- duces additional complexity due to the interplay between the matrices A1, A2, A3 and the diagonal structure of P1, P2, P3. 2. Background To establish a foundation for our results, we introduce several key concepts. All ma- trices considered in this work are real. Additionally, a matrix is positive (negative, respec- tively) definite if all its eigenvalues are positive (negative, respectively); meanwhile, we say it is positive (negative, respectively) semidefinite if all its eigenvalues are nonnegative (nonpositive, respectively). We shall adopt the notation X ≻ 0 (X ≺ 0, respectively) to indicate that a matrix X ∈ Rn×n is positive definite (negative definite, respectively). Similarly, we denote a positive semidefinite (negative semidefinite, respectively) matrix by X ⪰ 0 (X ⪯ 0, respectively). Unless stated otherwise, a positive or negative definite or semidefinite matrix is assumed to be symmetric. If X is a diagonal positive definite A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 3 of 10 matrix, we write positive diagonal matrix since it is clear that a diagonal positive definite matrix has all its diagonal elements positive. Let X ∈ Rn×n and u, v ∈ Rn. The trace of X is denoted by tr(X), and diag(X) represents the vector in Rn whose j-th component corresponds to the j-th diagonal element of X. We shall employ the notation u ≥ v to indicate that uj ≥ vj for all j = 1, . . . , n, and we use u ⩾ v if uj ⩾ vj for all j = 1, . . . , n. We denote the Hadamard product of two matrices X and Y , both in Rn×n, as X ◦ Y , where (X ◦ Y )ij = xijyij . A matrix X ∈ Rn×n is a P -matrix if every principal minor of X is positive. It is a well-established result (see Theorem 6.2.3 of [11]) that this condition is equivalent to the requirement that, for every non-zero u ∈ Rn, there exists an index i such that ui(Xu)i > 0. For a block matrix H ∈ R3n×3n, partitioned as H = H11 H12 H13 HT 12 H22 H23 HT 13 HT 23 H33  , we assume each block Hij ∈ Rn×n, and denote the (i, j)-th entry of H11 as h11ij = (H11)ij . Our analysis builds on several foundational results. For instance, it is well-known that the Hadamard product of two positive semidefinite matrices is positive semidefinite, and if both are positive definite, their product remains positive definite. Furthermore, if a matrix is Lyapunov diagonally stable—meaning there exists a positive diagonal matrix P such that ATP + PA is negative definite—then its negative is a P -matrix. These properties underpin our exploration of diagonal solutions and their stability characteristics. Lemma 1 ([16, 17]). Suppose A and B are n × n positive semidefinite matrices. Then the Hadamard product A ◦B is also positive semidefinite. Moreover, if A and B are both positive definite, their Hadamard product A ◦B will be positive definite. Lemma 2 ([10, 12]). If a matrix A ∈ Rn×n is Lyapunov diagonally stable, then −A is a P -matrix. Let A1, A2, A3 ∈ Rn×n. We say that the triple (A1, A2, A3) admits a positive diagonal solution if there exist positive diagonal matrices P1, P2, P3 ∈ Rn×n such that the block matrix AT 1 P1 + P1A1 + P2 + P3 P1A2 P1A3 AT 2 P1 −P2 0 AT 3 P1 0 −P3  (1) is negative definite. Furthermore, we say that (P1, P2, P3) is a positive diagonal solution for the triple (A1, A2, A3). Lemma 3 ([18]). Let A1, A2, and A3 be real n × n matrices. If the triple (A1, A2, A3) admits a positive diagonal solution, then A1 + A2 + A3 is a Lyapunov diagonally stable matrix. A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 4 of 10 3. Main Results This section presents our key findings on positive diagonal solutions for the LMI de- fined by the triple (A1, A2, A3) and the block matrix B. Extending classical stability criteria, we establish equivalent conditions linking B negative definiteness to test matrices and P -matrix properties under Hadamard transformations. We begin with preliminary equivalences (Theorems 2.1 and 2.2), followed by new characterizations (Theorems 2.3 and 2.4), offering tools for stability analysis in multi-matrix systems. 3.1. Preliminaries Theorem 1 ([18]). Let A1, A2, A3 ∈ Rn×n. Then, the following statements are equivalent: (i) There are positive diagonal matrices P1, P2, and P3 such that B = AT 1 P1 + P1A1 + P2 + P3 P1A2 P1A3 AT 2 P1 −P2 0 AT 3 P1 0 −P3  ≺ 0. (2) (ii) For any nonzero positive semidefinite matrix H ∈ R3n×3n, partitioned into 3 × 3 block matrices with each block in Rn×n as H = H11 H12 H13 HT 12 H22 H23 HT 13 HT 23 H33  , (3) satisfying the conditions diag(H11) ≥ diag(H22) and diag(H11) ≥ diag(H33), at least one diagonal element of the matrix A1H11 +A2H T 12 +A3H T 13 is negative. Theorem 2 ([18]). Let A1, A2, A3 ∈ Rn×n. Then, the following statements are equivalent: (i) There are positive diagonal matrices P1, P2, and P3 such that B = AT 1 P1 + P1A1 + P2 + P3 P1A2 P1A3 AT 2 P1 −P2 0 AT 3 P1 0 −P3  ≺ 0. (ii) For any nonzero positive semidefinite matrix H ∈ R3n×3n, partitioned into 3 × 3 block matrices with each block in Rn×n as H = H11 H12 H13 HT 12 H22 H23 HT 13 HT 23 H33  , A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 5 of 10 satisfying the conditions diag(H11) = diag(H22) = diag(H33), at least one diagonal element of the matrix A1H11 +A2H T 12 +A3H T 13 is negative. 3.2. New characterizations Theorem 3. Let A1, A2, A3 ∈ Rn×n. The triple (A1, A2, A3) admits a positive diagonal solution with respect to (1) if and only if the triple (A1 ◦ S11, A2 ◦ S12, A3 ◦ S13) also admits a positive diagonal solution with respect to (1) for any positive semidefinite matrix S ∈ R3n×3n of the form S = S11 S12 S13 ST 12 S22 S23 ST 13 ST 23 S33  , where S11, S22, S33 satisfy the condition diag(S11) = diag(S22) = diag(S33) ≫ 0. Proof. Necessity: Suppose that H is a nonzero positive semidefinite matrix in R3n×3n in the form H = H11 H12 H13 HT 12 H22 H23 HT 13 HT 23 H33  , with diag(H11) = diag(H22) = diag(H33). Now, define the matrix K = S ◦H. According to Lemma 1, K is a positive semidefinite matrix. Furthermore, as the diagonal entries of S are strictly positive and H is a nonzero matrix, then K must be a nonzero matrix. Additionally, by the conditions diag(H11) = diag(H22) = diag(H33) and diag(S11) = diag(S22) = diag(S33), it is clear that the same condition holds for K as well, i.e., diag(K11) = diag(K22) = diag(K33). Since, by assumption, the triple (A1, A2, A3) admits a positive diagonal solution with respect to (1), then, by Theorem 1, the matrix A1K11 +A2K T 12 +A3K T 13 A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 6 of 10 has at least one diagonal element that is less than zero. Next, observe thatK11 = S11◦H11, K12 = S12 ◦H12, and K13 = S13 ◦H13. Moreover, for every k ∈ {1, . . . , n}, we have (A1K11)kk = (A1(S11 ◦H11))kk = n∑ i=1 a1ki(S11 ◦H11)ik = n∑ i=1 a1kis 11 ikh 11 ik = n∑ i=1 a1kis 11 kih 11 ik = n∑ i=1 (A1 ◦ S11)kih 11 ik = ((A1 ◦ S11)H11)kk. Similarly, we have (A2K T 12)kk = (A2(S T 12 ◦HT 12))kk = n∑ i=1 a2ki(S T 12 ◦HT 12)ik = n∑ i=1 a2kis 12 kih 12 ki = n∑ i=1 (A2 ◦ S12)kih 12 ki = ((A2 ◦ S12)H T 12)kk, and using the same computations, we obtain that (A3K T 13)kk = ((A3 ◦ S13)H T 13)kk. Since A1K11 +A2K T 12 +A3K T 13 has at least one diagonal element, i.e., there is some k ∈ {1, . . . , n} such that (A1K11 +A2K T 12 +A3K T 13)kk < 0, hence, it follows that ((A1 ◦ S11)H11)kk + ((A2 ◦ S12)H T 12)kk + ((A3 ◦ S13)H T 13)kk < 0. Finally, using Theorem 2, this implies that the triple (A1 ◦ S11, A2 ◦ S12, A3 ◦ S13) admits a positive diagonal solution with respect to (1). A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 7 of 10 Sufficiency: Let S be the 3n×3n matrix of all ones, which is positive semidefinite with diag(S11) = diag(S22) = diag(S33) = 1 ≫ 0. Then, for each i, Ai ◦ Sii = Ai, and the condition reduces to the original triple (A1, A2, A3), which holds by assumption. Now, the proof is complete. Theorem 4. Let A1, A2, A3 ∈ Rn×n. The triple (A1, A2, A3) admits a positive diagonal solution with respect to (1) if and only if −(A1 ◦ S11 +A2 ◦ S12 +A3 ◦ S13) is a P -matrix for any positive semidefinite matrix S ∈ R3n×3n of the form S = S11 S12 S13 ST 12 S22 S23 ST 13 ST 23 S33  , where S11, S22, S33 satisfy the condition diag(S11) = diag(S22) = diag(S33) ≫ 0. Proof. Necessity: By assuming that A1, A2, A3 ∈ Rn×n admits a positive diagonal solution, it follows, by Theorem 3, that the triple (A1 ◦ S11, A2 ◦ S12, A3 ◦ S13) has a positive diagonal solution as well. Thus, by Lemma 3, the matrix A1 ◦ S11 +A2 ◦ S12 +A3 ◦ S13 is Lyapunov diagonally stable. Finally, according to Lemma 2, this means that −(A1 ◦ S11 +A2 ◦ S12 +A3 ◦ S13) is a P -matrix. Sufficiency: Suppose that H is a nonzero positive semidefinite matrix in R3n×3n parti- tioned as the following H = H11 H12 H13 HT 12 H22 H23 HT 13 HT 23 H33  , and satisfying the condition diag(H11) = diag(H22) = diag(H33). Next, construct a matrix S ∈ R3n×3n to be such that sij =  hij if i ̸= j hii if i = j and hii > 0 1 if i = j and hii = 0. Clearly, S is positive semidefinite with strictly positive diagonal entries. Furthermore, it satisfies the condition diag(S11) = diag(S22) = diag(S33) because diag(H11) = diag(H22) = diag(H33). Therefore, we have −(A1 ◦ S11 +A2 ◦ S12 +A3 ◦ S13) A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 8 of 10 is a P -matrix. Now, construct v ∈ Rn to be such that vk = 1 when hkk > 0 and vk = 0 when hkk = 0 for k ∈ {1, . . . , n}. Hence, using the P -matrix properties, there is an index k such that vk[(A1 ◦ S11 +A2 ◦ S12 +A3 ◦ S13)v]k < 0. This inequality means that n∑ i=1 a1kih 11 ki + n∑ i=1 a2kih 12 ki + n∑ i=1 a3kih 13 ki < 0, which implies that (A1H11 +A2H T 12 +A3H T 13)kk < 0. It follows, from Theorem 2, that the triple (A1, A2, A3) admits a positive diagonal solution. Now, the proof is complete. 4. Conclusions We have characterized the existence of positive diagonal solutions for a class of LMIs, providing equivalent conditions via test matrices (Theorems 1, 2) and Hadamard product transformations (Theorems 3, 4). These results generalize classical stability criteria and offer tools for analyzing multi-matrix systems in control and dynamics. 5. Conclusions In this paper, we have investigated the existence of positive diagonal solutions for a class of linear matrix inequalities involving a structured block matrix defined by a triple of matrices (A1, A2, A3). We established several equivalent conditions that characterize such solutions, linking the negative definiteness of the block matrix to properties of positive semidefinite test matrices and to the P-matrix property of transformed matrices under Hadamard products. These results generalize classical Lyapunov diagonal stability conditions to a multi- matrix setting, offering new insights and tools for the stability analysis of complex systems. In particular, the characterizations provided in this paper contribute to the understand- ing of how structural constraints and matrix interactions affect stability, with potential applications in control theory, networked systems, and biological models. Future work may explore numerical algorithms to compute such diagonal solutions efficiently and investigate extensions to nonlinear systems or systems with time-varying parameters. The approach developed here opens avenues for further research on structured stability criteria in high-dimensional and interconnected dynamical systems. A. Algefary, T. Alhumaidan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6417 9 of 10 Acknowledgements The authors gratefully acknowledge Qassim University, represented by the Deanship of Graduate Studies and Scientific Research, on the financial support for this research under the number (QU-J-UG-2-2025-56237) during the academic year 1446 AH / 2024 AD. We also sincerely thank the reviewers for their valuable feedback and constructive comments, which have significantly contributed to enhancing the quality of this paper. References [1] Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan. Linear matrix inequalities in system and control theory. SIAM, 1994. [2] Pascal Gahinet and Pierre Apkarian. A linear matrix inequality approach to h∞ control. 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