Hermitian solutions of the equation X = Q + NX−1N∗
Hermitian solutions of the equation X = Q + NX−1N∗
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DOI：
10.1016/0024-3795(95)00121-2
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发表时间：
1996-11
影响因子：
1.1
通讯作者：
A. Ferrante;B. Levy
中科院分区：
文献类型：
--
作者：
A. Ferrante;B. Levy
我们考虑矩阵方程X = Q + NX− 1 N <$。它的厄米特解被参数化为某个矩阵束的广义拉格朗日特征空间。我们表明，该方程承认最大和最小的解决方案。最大解对应于唯一的正定解。最小解是唯一负定解当且仅当N是非奇异的。如果N是奇异的，则不存在负定解。文中还得到了该方程与卡尔曼滤波理论中的标准代数Riccati方程之间的一个有趣关系。最后，我们提出了一个算法，收敛到正定解的初始条件范围很广。
We consider the matrix equation X = Q + NX−1N∗ . Its Hermitian solutions are parametrized in terms of the generalized Lagrangian eigenspaces of a certain matrix pencil. We show that the equation admits both a largest and a smallest solution. The largest solution corresponds to the unique positive definite solution. The smallest solution is the unique negative definite solution if and only if N is nonsingular. If N is singular, no negative definite solution exists. An interesting relation between the given equation and a standard algebraic Riccati equation of Kalman filtering theory is also obtained. Finally, we present an algorithm which converges to the positive definite solution for a wide range of initial conditions.