基于反Krylov矩阵正交分解的半可分矩阵

摘要

In this paper we prove that the orthogonal factor appearing in the QR factorization of the inserted Krylov matrix transforms a symmetric matrix having pairwise distinct eigenvalues into a semiseparable form.The result shows the relationship between the semiseparable matrices and the inverted Krylov matrix. In addition, we prove that the set of symmetric semiseparable matrices are invariant under QR iterations.%在本文中,我们证明了对一个反Krylov矩阵作QR分解后,利用得到的正交矩阵可以将一个具有互异特征值的对称矩阵转化为一个半可分矩阵的形式,这个结粜表明了反Krylov矩阵与半可分矩阵之间的联系.另外,我们还证明了这类对称半可分矩阵在QR选代下矩阼结构保持不变性.