最小多项式(minimalpolynomial)是代数数论的基本概念之一。由Cayley-Hamilton定理，A的特征多项式是A的零化多项式，而在A的零化多项式中，次数最低的首一多项式称为A的最小多项式。
最小多项式的求解方法
方法：
1、先将A的特征多项式
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在P中作标准分解，找到A的全部特征值
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的因式按次数从低到高的顺序进行检测，第一个能零化A的多项式就是最小多项式。
例：
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的最小多项式。
解：A的特征多项式为：
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又
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故A的最小多项式为
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扩展资料
特征多项式的解法
1、把|λE-A|的各行（或各列）加起来，若相等，则把相等的部分提出来（一次因式）后，剩下的部分是二次多项式，肯定可以分解因式。
2、把|λE-A|的某一行（或某一列）中不含λ的两个元素之一化为零，往往会出现公因子，提出来，剩下的又是一二次多项式。
3、试根法分解因式。
根据特征矩阵xE-A的标准形，其中次数最高的不变因子就是最小多项式
根据Jordan标准形
求次数最小的首项系数为1的零化多项式
根据有理标准型
初等因子等