论文标题
特征值的对称多项式的魔术决定因素公式
A magic determinant formula for symmetric polynomials of eigenvalues
论文作者
论文摘要
Symmetric polynomials of the roots of a polynomial can be written as polynomials of the coefficients, and by applying this to the characteristic polynomial we can write a symmetric polynomial of the eigenvalues $a_{i}$ of an $n\times n$ matrix $A$ as a polynomial of the entries of the matrix.我们为此提供了一个魔术公式：在对称的多项式中象征性地替换$ a \ mapsto a $，然后用$ \ det $替换乘法。例如，对于$ 2 \ times2 $矩阵$ a $，带有eigenvalues $ a_ {1}，a_ {2} $，\ begin \ begin {align*} a_1 a_1 a_1 a_1 a_1^2 + a_1^2 a_2 a_2 a_2＆= \ det（a_1，a_2^2），a_2^2） $ a_i^k $是$ i $ -th列的$ a^k $。人们也可以采用负功率，允许我们计算：\ begin {align*} a_1a_2^{ - 1}+a_1^{ - 1} a_ {2} a_ {2}＆= \ det（a_ {1}，a_ {1}，a_ {2}^{2}^{ - 1}^{ - 1} \ end {align*}魔术方法还适用于一组通勤矩阵的特征值的多元对称多项式，例如$ 2 \ times2 $矩阵$ a $和$ b $，带有特征$ a_1，a_2 $和$ b_ {1}，b_ {2} $，\ begin \ begin {align*} a_1 b_1 a_1 a_1 a_2 a_2^2^2 + a_1^2 + a_1^2a_2b_2b_2＆= \ \ \ \ \ \ \ \ \ ab_________________2 \ det（a_1^2，ab_2）\ end {align*}
Symmetric polynomials of the roots of a polynomial can be written as polynomials of the coefficients, and by applying this to the characteristic polynomial we can write a symmetric polynomial of the eigenvalues $a_{i}$ of an $n\times n$ matrix $A$ as a polynomial of the entries of the matrix. We give a magic formula for this: symbolically substitute $a\mapsto A$ in the symmetric polynomial and replace multiplication by $\det$. For instance, for a $2\times2$ matrix $A$ with eigenvalues $a_{1},a_{2}$, \begin{align*} a_1 a_2^2 +a_1^2 a_2 & =\det(A_1, A_2^2)+ \det(A_1^2, A_2) \end{align*} where $A_i^k$ is the $i$-th column of $A^k$. One may also take negative powers, allowing us to calculate: \begin{align*} a_1a_2^{-1}+a_1^{-1}a_{2} & =\det(A_{1},A_{2}^{-1})+\det(A_1^{-1},A_{2}) \end{align*} The magic method also works for multivariate symmetric polynomials of the eigenvalues of a set of commuting matrices, e.g. for $2\times2$ matrices $A$ and $B$ with eigenvalues $a_1,a_2$ and $b_{1},b_{2}$, \begin{align*} a_1 b_1 a_2^2 + a_1^2a_2b_2 & = \det(AB_{1},A_2^2) + \det(A_1^2,AB_2) \end{align*}
