关键词: 矩阵对角化, 多项式理论, 斐波那契数列, 通项公式, 二阶线性递推序列

Abstract: Based on matrix diagonalization and polynomial theory, a new proof of the general term formula of Fibonacci sequence is proposed and extended to the general second-order linear recursive sequence. By establishing the matrix representation of the recursive relationship, the matrix power is linearly expressed by using the minimum polynomial and eigenvalue theory of the matrix, and then the general term formula is derived. This method avoids the tedious steps of inverting the eigenvector matrix in the traditional diagonalization process, simplifies the derivation process and reflects the effectiveness of polynomial theory in the construction of matrix functions. Furthermore, the obtained results are extended to the general second-order linear recursive sequence, and a unified general term formula expression is established, which expands the scope of application of the method and has certain theoretical value and teaching reference significance.

Key words: matrix diagonalization, polynomial theory, Fibonacci sequence, general term formula, second-order linear recurrence sequence