二阶齐次线性递推数列两种解法的几何统一性探究

Exploration of the Geometric Unity of Two Solution Methods for Second-Order Homogeneous Linear Recursive Sequences

摘要: 本文构造投影映射π，将相空间向量映射为分量比形式的射影坐标，证明了友矩阵特征向量的投影即为不动点，从几何视角统一了二阶齐次线性递推数列的两类经典解法：不动点法与线性代数法。两种方法的对角化过程在该映射下高度对应，并针对特征值重根情形，明确两种方法分别对应若尔当标准型与等差数列结构，完善了理论框架，同时借助兰彻斯特方程给出了解释，为大学数学的学习和研究提供理论参考。

Abstract: This paper constructs a mapping π that projects phase space vectors to projective coordinates in the form of component ratios. It proves that the projection of eigenvectors of the companion matrix corresponds to fixed points, unifying the fixed-point method and linear algebra method for second-order homogeneous linear recursive sequences from a geometric perspective. Their diagonalization processes correspond highly under this mapping. For repeated eigenvalues, the two methods correspond to the Jordan canonical form and arithmetic progression structure, completing the theory. An explanation using the Lanchester equation is also given, providing a theoretical reference for advanced mathematics learning and research.