Logarithmic derivates of a square matrix
Logarithmic derivates of a square matrix
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DOI：
10.1016/0024-3795(73)90015-3
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发表时间：
1973
影响因子：
1.1
通讯作者：
C. Pao
中科院分区：
文献类型：
--
作者：
C. Pao
众所周知，n×n方阵A的对数导数μ[A]的值是关于任意范数的max Re(λi)从下有界的，其中λi(i=1，2，…，n)是OFA的特征值。在微分方程组的稳定性理论中，需要知道μ[A]的最小值。这篇注记的目的是给出一种构造算子范数的直接方法，使μ[A]关于它可以任意逼近max Re(λi)，并证明了一般的μ[A]≠max Re(λi)。给出了非线性摄动下线性方程稳定性问题的一个应用。
It is well-known that the value of the logarithmic derivative μ[A] of ann×nsquare matrixAis bounded from below by max Re(λi) with respect to any norm ofA, where λi(i= 1, 2,…,n) are the eigenvalues ofA. In the stability theory of differential equations it is desirable to know the smallest value of μ[A]. The purpose of this note is to give a direct method for the construction of an operator norm with respect to which μ[A] can be made arbitrarily close to max Re(λi) and to show that in general μ[A] ≠ max Re(λi). An application is given to the stability problem of a linear equation under nonlinear perturbations.