特征值与特征向量计算的非行列式方法

Computing Eigenvalues and Eigenvectors without Determinants

矩阵特征值和特征向量的计算问题在代数学中具有重要意义.传统教科书和相关文献给出的方法最终都要归结为求特征多项式的根,因此这些方法总是离不开行列式.基于行列式的特征值算法的最大缺点在于,当矩阵的阶数增大时,从行列式的表达式到其标准式,往往需要耗费大量的计算.为了避免使用行列式,探讨矩阵特征值与特征向量计算的非行列式方法就显得非常必要.从实际计算的角度看,虽然这种方法未必是最优的,但它对于扎实掌握矩阵特征分析理论具有很大益处.

Computing the eigenvalues and eigenvectors are important problems in linear algebra. Becausethese methods which in traditional textbooks and the relevant literature were attributed to calculate the roots of a characteristic polynomial, therefore,these methods always depended on the determinant. The algorithm of eigenvalue that based on the aeterminant was imperfect. The reason is when the matrix order increases, obtains the standard formula from the determinant needs to spend more calculation. In order to avoid using determinant, we explore the method to compute eigenvalues and eigenvectors without determinants is very necessary.From the computational view,although this method may not be the best,but it is useful to grasp matrix eigenvalue analysis theory.