Positive and completely positive cones and Z-transformations
Positive and completely positive cones and Z-transformations
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正锥体和完全正锥体以及 Z 变换
DOI：
10.13001/1081-3810.1515
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发表时间：
2012
影响因子：
0.7
通讯作者：
M. Gowda
中科院分区：
文献类型：
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作者：
M. Gowda
李雅普诺夫关于连续线性系统的一个著名结果是：一个真实的方阵A是正稳定的当且仅当对某个对称正定矩阵X，AX + XAT也是正定矩阵. Moldovan-Gowda最近的一个结果是：Z-矩阵A是正稳定的当且仅当对某个对称严格共正矩阵X，AX + XAT也是严格共正的.在本文中，我们通过用一个满足C = Rn的闭凸锥C代替Rn和Rn，统一/推广了这些结果。这是通过将此锥上矩阵的Z-性质与C的完全正锥上相应的李雅普诺夫变换LA（X）：= AX + XAT的Z-性质以及Sn（所有真实的nn对称矩阵的空间）中C的共正锥上LAT的Z-性质相联系来实现的.我们对Stein变换SA（X）= X AXA T进行类似的分析：
A well-known result of Lyapunov on continuous linear systems asserts that a real square matrix A is positive stable if and only if for some symmetric positive denite matrix X, AX + XA T is also positive denite. A recent result of Moldovan-Gowda says that a Z-matrix A is positive stable if and only if for some symmetric strictly copositive matrix X, AX + XA T is also strictly copositive. In this paper, we unify/extend these results by replacing R n and R n by a closed convex coneC satisfyingC C = Rn. This is achieved by relating the Z-property of a matrix on this cone with the Z-property of the corresponding Lyapunov transformation LA(X) := AX +XA T on the completely positive cone of C and the Z-property of L AT on the copositive cone of C in S n (the space of all real n n symmetric matrices). We carry out a similar analysis for the Stein transformation SA(X) = X AXA T :