## LaTeX2WP, Princeton grad student seminar, and characteristic polynomial coefficientsNovember 15, 2009

Posted by Phi. Isett in Uncategorized.

(It’ll be pretty cool if it catches on! But for me it probably means I will help to run the blog, and for this purpose I’ll definitely need the converter.)

I need to at least try to write some kind of math, so I’ll explain something which I think is cute: how to express the coefficients of a characteristic polynomial of a matrix in terms of sums of determinants of other matrices constructed from its entries.  Actually, I’ll first give an example which contains all the ideas. Consider the ${3 \times 3}$ matrix ${A}$ whose entries are… Let’s say

$\displaystyle A = \left( \begin{array}{ccc} 1 & 2 & 5 \\ 3 & 4 & 7 \\ 6 & 8 & 9 \end{array} \right)$

The characteristic polynomial ${\chi(x)}$ is the determinant of the matrix ${xI - A}$ where $I$ is the ${3 \times 3}$ identity matrix. ${\chi(x)}$ is a degree 3 polynomial with a leading coefficient 1. In terms of the ${\Lambda^3({\mathbb R}^3)}$, we have

$\displaystyle \begin{array}{c} x-1 \\ -3 \\ -6 \end{array} \wedge \begin{array}{c} -2 \\ x-4 \\ -8 \end{array} \wedge \begin{array}{c} -5 \\ -7 \\ x-9 \end{array} = \chi(x)\cdot e_1 \wedge e_2 \wedge e_3 . \ \ \ \ \ (1)$

I discussed the one-dimensional vector space ${\Lambda^3({\mathbb R}^3)}$ and its geometric meaning in a previous post about the Cauchy-Schwartz inequality and integration. We know that ${\chi(x) = x^3 + c_2 x^2 + c_1 x + c_0}$, where ${c_0 = \chi(0)}$, and plugging in ${0}$ into the above expression, we see that ${c_0 = (-1)^3 \det(A) = \det(-A)}$. The point of this entry is that we can calculate the other derivatives by differentiating, and use the multi-linearity of the wedge product to differentiate easily.  Below the fold I will give an example of how this computation works out, I will state what nice, general formula is proven by this method, and I will discuss the geometric meaning of this computation.  (And at the end I will ask a question about this LaTeX2WP/Python business which is still troubling me)