Timeline for Canonical form of symmetric integer matrix M
Current License: CC BY-SA 3.0
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| May 20, 2012 at 22:29 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| May 20, 2012 at 19:48 | comment | added | Will Jagy | @Xiao-Gang Wen, On the other hand, the Hahn survey is very nice and the theorem you quote entirely clear. Strange that I do not recognize his name, I guess he does the more algebraic side of this area. The references in his article are just about everything I would recommend for integral forms, outside of Watson and SPLAG. | |
| May 20, 2012 at 19:24 | comment | added | Will Jagy | @Xiao-Gang Wen, I think we need to allow for some of both types of 2 by 2 blocks. Given letters $w,x,y,z,$ I think the form $$ w^2 - x^2 + 2 y z $$ is not necessarily equivalent to any $$ r^2 - s^2 + t^2 - u^2 $$ in letters $r,s,t,u.$ This is what I was worried about when I wrote, above, "so the only problem is that $g$ may be even." Anyway, it should not take long to check this one example, now that I know what I am looking at. | |
| May 20, 2012 at 12:45 | comment | added | Xiao-Gang Wen | With the additional condition $JMJ^T=−M$, we see that the canonical form for even $M$ is the direct sum of $2x_1x_2$, and the canonical form for odd $M$ is the direct sum of $x_1^2−x_2^2$. | |
| May 20, 2012 at 12:43 | comment | added | Xiao-Gang Wen | Indeed, one needs the right refs (and right term to Google). On page 672 of math.nd.edu/assets/20630/hahntoulouse.pdf: An odd indefinite unimodular quadratic form over Z is equivalent to $M=a_1x_1^2+...+a_nx_n^2$ with $a_i=\pm 1$. An even indefinite unimodular quadratic form is equivalent to $M=2x_1x_2+2x_3x_4+...+E_8+E_8+...$ or $M=2x_1x_2+2x_3x_4+...-E_8-E_8-...$ where $E_8$ is the quadratic form of E8 lattice of the remaining variables. | |
| May 20, 2012 at 12:12 | vote | accept | Xiao-Gang Wen | ||
| May 20, 2012 at 5:00 | history | answered | Will Jagy | CC BY-SA 3.0 |