Timeline for What is the number of $2n\times 2n$-matrices $g=\begin{pmatrix} A & B \\\ C & -A^{t} \end{pmatrix}$, $B$ and $C$ symmetric, over the finite field $\mathbb{F}\_{q}$ with $\mathrm{rank}(g)=k$?

8 events

when toggle format	what		by	license	comment
Nov 8, 2012 at 7:49	vote	accept	micha
Nov 7, 2012 at 18:00	answer	added	Gerhard Paseman		timeline score: 3
Nov 6, 2012 at 16:56	comment	added	Gerhard Paseman		Try the following: look at the map which takes a matrix of your form to D=[B,A,,A^t,-C]. Ideally the map is bijective and rank preserving. Now if I have not messed up, your problem reduces to counting order 2n matrices which are of rank k and symmetric, for which you may find assistance in the literature. Gerhard "Check All These Suggestions Carefully" Paseman, 2012.11.06
Nov 6, 2012 at 15:05	comment	added	micha		I need the above as a lemma to deduce some deaper results about distribution of special matrices over finite rings.
Nov 6, 2012 at 15:00	history	edited	micha	CC BY-SA 3.0	deleted 16 characters in body
Oct 25, 2012 at 8:11	history	edited	micha	CC BY-SA 3.0	added 381 characters in body
Oct 24, 2012 at 13:37	comment	added	Gerhard Paseman		If k is odd, I think that will give very strict conditions on A. You might consider asking the question for k=1 or k=3, and perhaps doing a computer enumeration leveraged by whatever thery you can derive. Gerhard "Ask Me About System Design" Paseman, 2012.10.24
Oct 24, 2012 at 12:11	history	asked	micha	CC BY-SA 3.0