论文标题
排名一个具有广义特征向量的扰动
Rank one perturbation with a generalized eigenvector
论文作者
论文摘要
在文献中已经大量研究了两个矩阵的约旦结构之间的关系，其中一个方形矩阵$ a $及其等级的一个更新的矩阵$ a+xb^*$具有特别的兴趣。 $ a+xb^*$的特征值，其中$ x $是$ a $ a $ a和$ b $的特征向量是一个任意矢量，是由Brauer在1952年以$ a $的特征表示的。特征向量。但是，在后一种情况下，对$ b $的限制进行了限制，以使更新的矩阵的光谱与$ a $相同。当$ x $是$ a+xb^*$的特征值和广义特征向量上，当$ x $是$ x $是广义的特征向量，而$ b $是任意的向量时，似乎没有结果。在本文中，我们表明，可以根据$ a $ $ a $和任意向量$ b $的广义特征向量涉及$ a $ $ a $的广义特征向量。
The relationship between the Jordan structures of two matrices sufficiently close has been largely studied in the literature, among which a square matrix $A$ and its rank one updated matrix of the form $A+xb^*$ are of special interest. The eigenvalues of $A+xb^*$, where $x$ is an eigenvector of $A$ and $b$ is an arbitrary vector, were first expressed in terms of eigenvalues of $A$ by Brauer in 1952. Jordan structures of $A$ and $A+xb^*$ have been studied, and similar results were obtained when a generalized eigenvector of $A$ was used instead of an eigenvector. However, in the latter case, restrictions on $b$ were put so that the spectrum of the updated matrix is the same as that of $A$. There does not seem to be results on the eigenvalues and generalized eigenvectors of $A+xb^*$ when $x$ is a generalized eigenvector and $b$ is an arbitrary vector. In this paper we show that the generalized eigenvectors of the updated matrix can be written in terms of those of $A$ when a generalized eigenvector of $A$ and an arbitrary vector $b$ are involved in the perturbation.
