On the existence of a positive definite solution of the matrix equation X+ATX−1A= I

Positive definite solutions of the nonlinear matrix equation X+AHX¯-1Zhou1, Cai2, Lam3 2013Applied Mathematics and Computation320190View full textAdd to dashboardCite

“…By applying Lemma 4, the above inequality is equivalent to (19). Assume that there exists another positive definite solution X * such that coρ X −1 + A ≤ 1.…”

“…Various kinds of matrix equations have received much attention in the literature (see, for example, [4], [7], [8], [11], [12], [13], [15], [14], [16], [19], [27], [28], [29], [34], [38], [39], [40], and the references therein). Especially, the problem of finding fixed points of the nonlinear matrix equation X + A * X −1 A = Q where A and Q > 0 are given and X is unknown, has been extensively studied in the last two decades.…”

“…Especially, the problem of finding fixed points of the nonlinear matrix equation X + A * X −1 A = Q where A and Q > 0 are given and X is unknown, has been extensively studied in the last two decades. The interest of studying such problem mainly relies on its applications in many fields such as the analysis of ladder networks [2], dynamic programming [33], control theory [22], stochastic filtering [1] and statistics [32] (see [3], [18] and [19] for detailed introduction). Some generalized forms of the nonlinear matrix equation X + A * X −1 A = Q have received much attention in recent years (see, for example, [6], [17], [24], and [30]).…”

“…The other kind of results are mainly concerned with the numerical solutions of this class of nonlinear matrix equations. Basically, this can be accomplished via iterations including inversion-involved iterations [19], [21] and inversion-free iterations [20], [23], [31], [36].…”

Iterative solution of two matrix equations

“…These two equations have been studied recently by several authors (see [1], [2], [3], [4], [16], [17]). For the application areas in which the equations arise, see the references given in [1] and [4].…”