数学归纳法在特征多项式方法中的应用

The Application of Mathematical Induction in the Method Based on the Characteristic Polynomials

本文利用数学归纳法证明sturm序列的性质——特征多项式的根是单根且相邻两个特征多项式的零点相互隔离。用不同于教材中的方法证明该数值代数中的已知结论,有助于引导本科生探索不同的思考方式,培养发散性思维能力。

This article uses mathematical induction to prove the property of sturm sequence that the zeros of the characteristic polynomials are simple and the zeros of two neighbouring characteristic poly-nomials are isolated from each other. Utilizing the method different from that in textbooks to prove this known conclusion in numerical algebra can help guide undergraduate students to explore different ways of thinking and cultivate divergent thinking abilities.