A magic determinant formula for symmetric polynomials of eigenvalues

Abstract

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*}