The Derivation of the Exponential Map of Matrices
The Derivation of the Exponential Map of Matrices
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矩阵指数图的推导
DOI：
10.1080/00029890.1995.12004668
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发表时间：
1995
影响因子：
0.5
通讯作者：
G. Tuynman
中科院分区：
文献类型：
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作者：
G. Tuynman
在[1]中概括了这个证明，似乎假设对于一个趋近于0的序列hk， m >n的商hm/htl是有界的。最后补充一点：Dini的散度检验表明，如果存在一个（弱）单调递增序列ck = mkak且El/Mk（£ak/ck）发散，则Eak是发散的。单调条件可以写成aJak+l 2 (l/mk)/(l/mk+l)，并通过一个使ak也发散的标准参数。（这是迪尼的证明。）让我们在这里用一个较弱的条件来代替ck的单调性，即ck的下界是一个正数，比如说1。然后我们有ak 2 ak/ck，后一项的级数通过假设发散，因此Dini准则的一个稍微改进的版本也被证明与比较检验相同（发散版本）！有趣的是，将ck写成mkak可以很容易地设置特定的测试
recapitulates this proof in [1], seems to assume that for a sequence hk that goes to O the quotients hm/htl for m > n are bounded. And as a last remark: Dini's test for divergence says that Eak is divergent if there exists a (weakly) monotone increasing sequence ck = mkak with El/Mk (£ak/ck) diverging. Ae monotonicit condition can be written as aJak+l 2 (l/mk)/(l/mk+l) and by a standard argument that makes Eak diverge also. (Ihat is Dini's proof.) Let us here replace the monotonicity of the ck by the weaker condition that the ck are bounded below by a positive number, say by 1. Then we have ak 2 ak/ck, with the series of the latter terms diverging by assumption, and thus a slightly improved version of Dini's criterion also turns out to be identical with the comparison test (divergence version)! The interesting aspect is again that writing the ck as mkak makes it easy to set up specific tests