论文标题
利用Cayley Hamilton定理有效解决Jordan规范形式问题
Leveraging the Cayley Hamilton Theorem for Efficiently Solving the Jordan Canonical Form Problem
论文作者
论文摘要
鉴于$ n \ times n $非源性矩阵A和A作为起点的特征多项式，我们将利用Cayley-Hamilton定理来有效地计算矩阵中每个独特的特征值的最大长度Jordan链。通过寻求某种类型的启动向量作为算法的第一步来获得效率和速度。找到此起始向量的方法不需要计算$ ker [（a-λi）^k] $，这是一个非常昂贵的操作，这是解决约旦规范基础问题的常规方法。鉴于此起始向量，约旦链中的所有剩余向量都是在循环中很快计算出来的。然后，包含约旦链的向量将用于最大程度地减少方程求解量，以找到其余的广义特征向量基础。我们将证明一个定理，证明了为什么由此产生的约旦连锁店最大长度。我们还将证明如何得出起始矢量，该矢量随后将用于计算最大的约旦链。
Given an $n \times n$ nonsingular matrix A and the characteristic polynomial of A as the starting point, we will leverage the Cayley-Hamilton Theorem to efficiently calculate the maximal length Jordan Chains for each distinct eigenvalue of the matrix. Efficiency and speed are gained by seeking a certain type of starting vector as the first step of the algorithm. The method for finding this starting vector does not require calculating the $ker[(A-λI)^k]$ which is quite an expensive operation, and which is the usual approach taken in solving the Jordan Canonical basis problem. Given this starting vector, all remaining vectors in the Jordan Chain are calculated very quickly in a loop. The vectors comprising the Jordan Chains will then be used to minimize the amount of equation solving in order to find the remaining generalized eigenvector basis. We will prove a theorem that justifies why the resulting Jordan Chains are of maximal length. We will also justify how to derive the starting vector which will subsequently be used to calculate the Maximal Jordan Chains.
