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LaTeX2WP, Princeton grad student seminar, and characteristic polynomial coefficients November 15, 2009

Posted by Phi. Isett in Uncategorized.

Alright, I have gotten through my general exam and now maybe I’ll be able to do some blogging and actually get this blog up to industry standard (adding links to other blogs on the sides, etc.). I am writing this entry in order to test out Luca Trevisan’s LaTeX2WP (LaTeX to WordPress) converter, which is described on his site and can be downloaded for free. (So this is my learn-how-to-use-the-program post). It will clearly be helpful for my own blogging purposes for me to understand this software, but I will also be helping initiate another blog which will certainly require it. At Princeton, the pure math students have this Graduate Student Seminar where every Thursday we all eat pizza while somebody gives a lecture on whatever he/she feels like talking about. It seemed like a good idea to make a blog to accompany the seminar so that a broader audience could be reached, so I’ll include a link when the blog comes into being.

(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)