arXiv:1810.05033v2 [math.RA] 9 May 2019 Lyapunov matrix equation A H X + XA + C = O r with A a Jordan matrix Dan Com˘anescu Department of Mathematics, West University of Timi¸soara Bd. V. Pˆ arvan, No 4, 300223 Timi¸soara, Romˆania dan.comanescu@e-uvt.ro Abstract A Lyapunov matrix equation can be converted, by using the Jordan decomposition theorem for matrices, into an equivalent Lyapunov matrix equation where the matrix is a Jordan matrix. The Lyapunov matrix equation with Jordan matrix can be reduced to a system of Sylvester-Lyapunov type matrix equations. We completely solve the Sylvester-Lyapunov type matrix equations cor- responding to the Jordan block matrices of the initial matrix. MSC: 15A21, 15A24, 15B57 Keywords: Lyapunov matrix equation, Jordan matrix, Hermitian matrix. 1 Introduction A Lyapunov matrix equation (or the continuous Lyapunov matrix equation) has the form A H X + XA + C = O r , (1.1) with A ∈ M r (C), C ∈ M r (C) and the unknown X ∈ M r (C). The homogeneous Lyapunov matrix equation is: A H X + XA = O r . (1.2) The Lyapunov matrix equation appeared, see [11] and [7], in the study of the dynamics of a linear differential equation, ˙ x = Ax, where A ∈ M r (C). Dynamical aspects, like stability, are studied using a square function V (x)= 1 2 x H X x, with X ∈ M r (C) a Hermitian matrix. The function V is a conserved quantity of the linear system if and only if X is a Hermitian solution of the homogeneous Lyapunov matrix equation (1.2). The global asymptotic stability of the equilibrium point 0 is assured by the existence of a Hermitian positive definite solution of the Lyapunov matrix equation (1.1), with C a Hermitian positive definite matrix. Lyapunov matrix equations appear also in the study of optimization of a cost function defined on a matrix manifold, see [1]. The Lyapunov matrix equation is a particular case of the Sylvester matrix equation AX + XB + C = O r , (1.3) with A ∈ M r (C), B ∈ M s (C), C ∈ M r×s (C) and the unknown X ∈ M r×s (C). There are numerous studies of this equation in the literature, see [6, 12, 13], and we find some representations of the solutions for the Sylvester matrix equation, see [10], but they are difficult to use when r and s are large numbers. We also find studies of generalized Sylvester matrix equation, see [4], or studies of other matrix equations which are derived from the Lyapunov matrix equation, 1