论文标题
查找特征向量：快速和非传统方法
Finding Eigenvectors: Fast and Nontraditional Approach
论文作者
论文摘要
对角度化矩阵$ a $，发现两个矩阵$ p $和$ d $，使得$ a = pdp^{ - 1} $，$ d $是对角矩阵需要两个步骤：首先找到特征值，然后找到记录的特征向量。我们表明，当具有频谱的对角度化矩阵时，我们不需要第二步，$ \ left |σ（a）\ right | \ leq 2 $，因为这些向量已经显示为$ \ textit {eigenmatrices} $的非零列，这是在这项工作中定义的术语。我们将其进一步概括为具有$ \左|σ（a）\右|> 2 $的矩阵，并证明特征向量位于互补特征值的特征值的圆柱空间中，这种方法无需使用经典的高斯 - juss-jordan of of Matrix的行消除。 我们介绍了两个主要结果，即$ \ textit {2-spectrum lemma} $和$ \ textit {eigenmatrix theorem} $。作为一个猜想，我们进一步概括了约旦规范形式，用于通过某些特征性重复倍数产生的新的广义特征向量。我们还提供了几种快捷方式公式，以找到不使用梯队形式的特征向量。在这项工作中讨论的方法可以用助记符“在邻居那里找到您的小狗”！争论，小狗是特征向量，而邻居是互补的特征。
Diagonalizing a matrix $A$, that is finding two matrices $P$ and $D$ such that $A = PDP^{-1}$ with $D$ being a diagonal matrix needs two steps: first find the eigenvalues and then find the corresponding eigenvectors. We show that we do not need the second step when diagonalizing matrices with a spectrum, $\left|σ(A)\right|\leq 2$ since those vectors already appear as nonzero columns of the $\textit{eigenmatrices}$, a term defined in this work. We further generalize this for matrices with $\left|σ(A)\right|> 2$ and show that eigenvectors lie in the column spaces of eigenmatrices of the complementary eigenvalues, an approach without using the classical Gauss-Jordan elimination of rows of a matrix. We introduce two major results, namely, the $\textit{2-Spectrum Lemma}$ and the $\textit{Eigenmatrix Theorem}$. As a conjecture, we further generalize the Jordan canonical forms for a new class of generalized eigenvectors that are produced by repeated multiples of certain eigenmatrices. We also provide several shortcut formulas to find eigenvectors that does not use echelon forms. The method discussed in this work may be summarized with the mnemonic "Find your puppy at your neighbors'!" argument, where puppy is the eigenvector and the neighbors are the complementary eigenmatrices.
