Structured conditioning of matrix functions
Structured conditioning of matrix functions
复制标题
矩阵函数的结构化调节
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Philip I. Davies
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文献类型：
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作者：
Philip I. Davies
现有的矩阵函数f（X）：Cn ×n → Cn ×n的条件理论不考虑矩阵X的结构。这个理论的一个扩展，其中当X的结构，所有的扰动X需要有相同的结构。两类结构矩阵被认为是，包括约旦代数J和李代数L与非退化双线性或sesquilinear形式的Rn或Cn。这些类的例子是对称矩阵、反对称矩阵、哈密尔顿矩阵和反对称哈密尔顿矩阵。为这两个类定义了结构化条件数。在一定的条件下的基础标积，明确表示的结构条件数。然后，非结构化和结构化的条件数之间的比较。当基础标积是半双线性形式时，证明了（i）X ∈ J的所有函数，（ii）X ∈ L的奇函数和偶函数的两个条件数的值之间没有差别.当底层标量积是双线性形式时，在所有这些情况下都不能保证相等。在不保证相等的情况下，得到了非结构化和结构化条件数之比的界限。
The existing theory of conditioning for matrix functions f (X): C n×n → C n×n does not cater for structure in the matrix X. An extension of this theory is presented in which when X has structure, all perturbations of X are required to have the same structure. Two classes of structured matrices are considered, those comprising the Jordan algebra J and the Lie algebra L associated with a nondegenerate bilinear or sesquilinear form on R n or C n . Examples of such classes are the symmetric, skew-symmetric, Hamiltonian and skew-Hamiltonian matrices. Structured condition numbers are defined for these two classes. Under certain conditions on the underlying scalar product, explicit representations are given for the structured condition numbers. Comparisons between the unstructured and structured condition numbers are then made. When the underlying scalar product is a sesquilinear form, it is shown that there is no difference between the values of the two condition numbers for (i) all functions of X ∈ J, and (ii) odd and even functions of X ∈ L. When the underlying scalar product is a bilinear form then equality is not guaranteed in all these cases. Where equality is not guaranteed, bounds are obtained for the ratio of the unstructured and structured condition numbers.