On the ★-Sylvester equation AX±X★ B★=C

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

“…We use the idea found in [36] to prove the result. According to Theorem 1, we have X ♥ + = W + , the maximal solution to equation (7). Notice that (15) On the other hand, by using Lemma 2, we can compute…”

“…Proof of Item 1 : By Theorem 1, W + = X ♥ + is the maximal solution of the nonlinear matrix equation (7). Then according to Lemma 7 in appendix, W + is the unique positive definite solution of equation (7) such that W + + λA ♦ is nonsingular for all λ ∈ {z : |z| < 1}.…”

“…Under the assumption of this corollary, W + = X ♥ + and W − = X ♥ − are respectively the maximal and minimal solution of equation (7). Therefore, we need only to show that W + and W − are in the form of…”

“…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.…”

The solution of the equationAX + BX⋆ = 0

“…All these problems have been addressed (and, essentially, solved) by different authors for the general -Sylvester equation or the homogeneous case AX + X B = 0 of this equation [1,2,6,7,12,14,18].…”