Question

最小多项式

几种方法能求最小多项式？

Answer 1

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。

Answer 2

最小多项式(minimal polynomial)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。

数域P上n级矩阵A与对角矩阵相似的充分必要条件为A的最小多项式是P上互素的一次因式的乘积。

推论：复数矩阵A与对角矩阵相似的充分必要条件是A的最小多项式没有重根。

扩展资料：

最小多项式的性质：

1、A的最小多项式是唯一的。

2、相似的方阵阵具有相同的最小多项式。

3、A的小多项式g（r）是它的特征多项式f（r）=lrE-Al的一个因式。

参考资料来源：

百度百科-最小多项式

Answer 3

1. 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

2. 根据Jordan标准形

3. 求次数最小的首项系数为1的零化多项式

4. 根据有理标准型

5. 初等因子等