A Chain Rule for Matrix Functions and Applications
A Chain Rule for Matrix Functions and Applications
复制标题
矩阵函数的链式法则及其应用
DOI：
10.1137/s0895479895283409
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
R. Mathias
中科院分区：
文献类型：
--
作者：
R. Mathias
设f$是一个不一定解析的函数，设A（t）$是一个依赖于参数t$的n × n矩阵族。给出了f（A（t））$的一阶和高阶导数存在的条件，以及将这些导数表示为f（B）$的子矩阵的公式，其中B$是一个较大的块Toeplitz矩阵。这种块矩阵表示的一阶导数的矩阵函数的条件估计的上下文中是有用的。这里给出的结果比文献中的结果略强，并且以相当简单的方式证明。
Let $f$ be a not necessarily analytic function and let $A(t)$ be a family of $n \times n$ matrices depending on the parameter $t$. Conditions for the existence of the first and higher derivatives of $f(A(t))$ are presented together with formulae that represent these derivatives as a submatrix of $f(B)$, where $B$ is a larger block Toeplitz matrix. This block matrix representation of the first derivative is shown to be useful in the context of condition estimation for matrix functions. The results presented here are slightly stronger than those in the literature and are proved in a considerably simpler way.