论文标题
不变子空间不变子空间的扰动
Perturbation of invariant subspaces for ill-conditioned eigensystem
论文作者
论文摘要
鉴于可对角矩阵$ a $，我们研究了其不变子空间的稳定性，当它的特征向量矩阵不足时。令$ \ MATHCAL {X} _1 $为$ a $的某个不变子空间，$ x_1 $是存储跨度$ \ MATHCAL {x} _1 $的矩阵。通常认为，当条件编号$κ_2（x_1）$变大时，相应的不变子空间$ \ MATHCAL {x} _1 $将变得不稳定。本文证明并非总是如此。具体来说，我们表明，仅$κ_2（x_1）$的增长不足以破坏稳定性。作为一个直接应用程序，我们的结果确保了$ a $越来越接近约旦表格时，人们仍然可以从嘈杂的数据稳定地估算其不变子空间。
Given a diagonalizable matrix $A$, we study the stability of its invariant subspaces when its matrix of eigenvectors is ill-conditioned. Let $\mathcal{X}_1$ be some invariant subspace of $A$ and $X_1$ be the matrix storing the right eigenvectors that spanned $\mathcal{X}_1$. It is generally believed that when the condition number $κ_2(X_1)$ gets large, the corresponding invariant subspace $\mathcal{X}_1$ will become unstable to perturbation. This paper proves that this is not always the case. Specifically, we show that the growth of $κ_2(X_1)$ alone is not enough to destroy the stability. As a direct application, our result ensures that when $A$ gets closer to a Jordan form, one may still estimate its invariant subspaces from the noisy data stably.
