🧮 特征多项式计算器Characteristic Polynomial Calculator

使用说明与常见问题How-to & FAQ

「特征多项式计算器」是什么？What is the Characteristic Polynomial Calculator tool?

用分数精确计算方阵的特征多项式 p(λ) = det(λI − A)（Faddeev–LeVerrier 算法），同时给出迹、行列式，以及所有有理数特征值。Compute the characteristic polynomial p(λ) = det(λI − A) of a square matrix exactly with fractions (Faddeev–LeVerrier algorithm), with the trace, determinant and any rational eigenvalues.

如何使用「特征多项式计算器」？How do I use the Characteristic Polynomial Calculator tool?

在「特征多项式计算器」中，填写上方字段等参数，结果会随输入实时计算并显示在下方。In the Characteristic Polynomial Calculator: Fill in the fields above; results are recalculated instantly as you type.

特征多项式怎么算？How is the characteristic polynomial computed?

p(λ) = det(λI − A)。这里用 Faddeev–LeVerrier 递推：M₀ = 0，c_n = 1，M_k = A·M_{k−1} + c_{n−k+1}·I，c_{n−k} = −tr(A·M_k) ÷ k。全程分数运算，结果精确。p(λ) = det(λI − A). The tool uses the Faddeev–LeVerrier recursion: M₀ = 0, c_n = 1, M_k = A·M_{k−1} + c_{n−k+1}·I, c_{n−k} = −tr(A·M_k) ÷ k, with exact fractions throughout.

为什么只列出有理数特征值？Why only rational eigenvalues?

特征值是 p(λ) = 0 的根。工具用有理根定理精确找出所有有理根（并给出重数）；无理根和复数根需要数值方法，这里不列出。Eigenvalues are the roots of p(λ). The tool finds every rational root exactly with the rational root theorem (with multiplicity); irrational and complex roots need numerical methods and are not listed.

和 det(A − λI) 有什么不同？How does it relate to det(A − λI)?

det(A − λI) = (−1)ⁿ det(λI − A)，两者根相同。n 为奇数时符号相反。det(A − λI) = (−1)ⁿ det(λI − A); the roots are the same, and the sign flips when n is odd.

「特征多项式计算器」免费吗？需要注册吗？Is the Characteristic Polynomial Calculator free? Do I need an account?

「特征多项式计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Characteristic Polynomial Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.