Is there a small skew Cayley transform with zero diagonal
Is there a small skew Cayley transform with zero diagonal
复制标题
是否存在对角线为零的小偏凯莱变换
DOI：
10.1016/j.laa.2005.08.027
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发表时间：
2006
影响因子：
1.1
通讯作者：
W. Kahan
中科院分区：
文献类型：
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作者：
W. Kahan
厄米矩阵H的特征向量是某个复酉矩阵Q的列。对于任何对角酉矩阵Ω， Q·Ω的列也是特征向量。在所有的Q·Ω中，至少有一个存在一个歪斜的hermite - Cayley变换S，其中（I+Q·Ω）−1·（I−Q·Ω）的对角线上只有0。为什么?证明是不明显的，正如进一步的观察，Ω也可以这样选择，这个S的任何元素的大小都不需要超过1。因此，当厄米矩阵H受到无穷小扰动时，很容易通过复特征向量的扰动来满足的似是而非的约束也可以被离散扰动所满足。但是如果H是实对称的，Q是实正交的，并且Ω被限制在±1的对角线上，那么是否至少有一个实偏对称的S必须没有大小大于1的元素，我们还不知道。
The eigenvectors of an Hermitian matrix H are the columns of some complex unitary matrix Q. For any diagonal unitary matrix Ω the columns of Q·Ω are eigenvectors too. Among all such Q·Ω at least one has a skew-Hermitian Cayley transform S≔(I+Q·Ω)−1·(I−Q·Ω) with just zeros on its diagonal. Why? The proof is unobvious, as is the further observation that Ω may also be so chosen that no element of this S need exceed 1 in magnitude. Thus, plausible constraints, easy to satisfy by perturbations of complex eigenvectors when Hermitian matrix H is perturbed infinitesimally, can be satisfied for discrete perturbations too. But if H is real symmetric, Q real orthogonal and Ω restricted to diagonals of ±1’s, then whether at least one real skew-symmetric S must have no element bigger than 1 in magnitude is not known yet.