Using the Khatri-Rao product, we presented new characterizations for the common Lyapunov diagonal stability for a family of real matrices $\mathcal{A}$. For special partitions $\alpha$, we used the notion of ${\mathcal{P}}^{\alpha }$-sets and common $\alpha$-scalar Lyapunov stability to formulate further characterizations. Furthermore, generalizations of these results to the common $\alpha$-scalar Lyapunov stability were developed. Our goal of this paper was to unify and enhance relevant work.

In this work, an explicit formula in terms of the Hadamard product was derived for the solutions to fractional differential Stein matrix equations (FDSMEs), assuming that the coefficient matrices are separately diagonalizable. To begin with, we gave a general formula using the Kronecker product notation, which does not need any diagonalizability assumption to show the existence and uniqueness of the solution. In the case of diagonalizable coefficient matrices, the matrix equation became decoupled scalar Caputo fractional differential equations. Solutions to these differential equations were given explicitly in terms of two-parameter Mittag–Leffler functions and put together using the Hadamard product. The results for integer-order differential equations are obtained as special cases. Finally, we give three examples.

In this manuscript, we formulated Hermite–Hadamard inequalities for convex functions by employing cotangent integrals. Additionally, we extended these Hermite–Hadamard inequalities to encompass cotangent integrals and give the application.

In this paper, we present a characterization of diagonal solutions for a class of linear matrix inequalities. We consider linear hybrid time-delay systems and explore the conditions under which these systems are positive and asymptotically stable. Specifically, we investigate the existence of positive diagonal solutions for a linear inequality when the system matrices are Metzler and nonnegative. Using various mathematical tools, including the Schur complement and separation theorems, we derive necessary and sufficient conditions for the stability of these systems. Our results extend existing stability criteria and provide new insights into the stability analysis of positive time-delay systems.

In this short note, we presented a number of alternative explicit necessary and sufficient conditions for diagonal stability along with a new proof of a well-known result in this regard.