Perturbation Bounds for the Polar Decomposition
Perturbation Bounds for the Polar Decomposition
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DOI：
10.1137/0614041
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发表时间：
1993-04
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影响因子：
--
通讯作者：
R. Mathias
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文献类型：
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作者：
R. Mathias
设$M_n（F）$表示域F上的矩阵空间。给定$A \in M_n（F）$定义$|一|\equiv（A^ * A）^{1/2} $和$U（A）\equiv A|一|^{-1} $假设A是非奇异的。设$\sigma_1（A）\geq \sigma_2（A）\geq \cdots \sigma _n（A）\geq 0$表示A的有序奇异值，得到了U（A + \Delta A）- U（A）$与A和$\Delta A$奇异值之间的优控制结果.特别地，本文证明了如果$A，\，\Delta A \in M_n（R）$和$\sigma _1（\Delta A）< \sigma _n（A）$，则对于任何酉不变范数$\| \cdot \|，U（A + \Delta A）- U（A）\| \leq 2 [ \sigma_{n - 1}（A）+ \sigma_{n}（A）]^{ - 1} \Delta A \|$.本文还讨论了酉Procrustes问题：$min A-UB：U in M_n（C），U^ * U = I $，其中A，B \in M_n（C）$，以及一个酉不变范数$\| cdot \|$.假设U是幺正的，并且$U^ * BA^ * $是.
Let $M_n ( F )$ denote the space of matrices over the field F. Given $A \in M_n ( F )$ define $| A | \equiv ( A^ * A )^{1/2} $ and $U( A ) \equiv A | A |^{ - 1} $ assuming A is nonsingular. Let $\sigma _1 ( A ) \geq \sigma _2 ( A ) \geq \cdots \sigma _n ( A ) \geq 0$ denote the ordered singular values of A.Majorization results are obtained relating the singular values of $U ( A + \Delta A ) - U ( A )$ and those of A and $\Delta A$. In particular, it is shown that if $A,\,\Delta A \in M_n ( R )$ and $\sigma _1 ( \Delta A ) < \sigma _n ( A )$, then for any unitarily invariant norm $\| \cdot \|, \| U ( A + \Delta A ) - U ( A ) \| \leq 2 [ \sigma_{n - 1} ( A ) + \sigma_{n} ( A )]^{ - 1} \| \Delta A \|$. Similar results are obtained for matrices with complex entries.Also considered is the unitary Procrustes problem: $\min \{ \| A - UB \|:U \in M_n ( C ),U^ * U = I \}$ where $A,B \in M_n ( C )$, and a unitarily invariant norm $\| \cdot \|$ are given. It was conjectured that if U is unitary and $U^ * BA^ * $ is ...