Timeline for Geometric proof of the Vandermonde determinant?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| May 31, 2014 at 6:20 | comment | added | Tom Copeland | S. Winitzki in Linear Algebra via Exterior Products (pages 126-7) gives a derivation of the product formula that seems to combine some of the observations of Algol and Terry Tao. | |
| May 19, 2014 at 1:23 | comment | added | Tom Copeland | Similarly, with $e_k(x_1,...,x_n)$ the elementary symmetric polynomials, $de_1 \wedge de_2 \; ...\wedge de_n= |V_n|\; dx_1 \wedge dx_2 \; ...\wedge dx_n.$ | |
| Oct 31, 2010 at 17:46 | comment | added | Terry Tao | One can phrase this a little bit more geometrically as follows. The $n$-dimensional space $V$ of polynomials of degree less than $n$ has two bases: the standard basis $1,x,\ldots,x^{n-1}$, and Agol's basis $1, (x-x_1), (x-x_1)(x-x_2),\ldots,(x-x_1)\ldots(x-x_n)$. It also has the map $\Phi: P \mapsto (P(x_1),\ldots,P(x_n))$ from $V$ to $R^n$. The Vandermonde det is det of $\Phi$ with respect to the standard basis, while the product $\prod_{1 \leq i < j \leq n} (x_i-x_j)$ is the det of $\Phi$ wrt Agol's basis (where it becomes triangular). But the two bases are linked by a unipotent map. | |
| Jun 25, 2010 at 16:35 | comment | added | Ian Agol | @Daniel: yes, if you look at the row reduction argument given in your link, it's actually what you get by doing only the column operations, which is how I found it (since I wanted to see what kind of affine transformation this produced). The matrix can be factored into block unipotent matrices which have $n−1−j$ copies of $−x_j$ just above the diagonal in the lower right corner, for $j=1,\ldots,n−1$. | |
| Jun 25, 2010 at 13:05 | comment | added | Daniel Litt | I think this is exactly the same as row reduction if you write it out, no? That said, it's a great way of doing the computation. | |
| Jun 25, 2010 at 4:41 | comment | added | Wadim Zudilin | It sounds like en.wikipedia.org/wiki/Companion_matrix. | |
| Jun 25, 2010 at 3:23 | history | made wiki | Post Made Community Wiki by Ian Agol | ||
| Jun 25, 2010 at 3:18 | history | answered | Ian Agol | CC BY-SA 2.5 |