On the definiteness of the solutions of certain matrix equations
On the definiteness of the solutions of certain matrix equations
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关于某些矩阵方程解的确定性
DOI：
10.1016/0024-3795(88)90187-5
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发表时间：
1988
影响因子：
1.1
通讯作者：
M. Kwong
中科院分区：
文献类型：
--
作者：
M. Kwong
证明了对任意维数n <$2，存在真实的tn使得对所有t <$2，tn]，矩阵方程A2 X + XA 2 +tAXA= B的n × n解X必是半正定的，对任意正定A和半正定B.这推广了经典的李雅普诺夫结果。对于n = 2，3，4，分别成立attn = ∞，8，4;对于所有n，成立attn> 2.给出了n到6的上界。并对有关方程进行了讨论。
It is shown that for each dimensionn⩾ 2, there is a real numbertnsuch that for alltϵ(−2,tn], the uniquen×nsolutionXof the matrix equationA2X+XA2+tAXA=Bmust be positive semidefinite, for any positive definiteAand positive semidefiniteB. This generalizes the classical Lyapunov result. It is established thattn= ∞, 8, 4 forn= 2, 3, 4 respectively and thattn> 2 for alln. Upper bounds are given fornup to 6. Related equations are also discussed.