Timeline for Determinant of "skew-symmetric" matrices
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2019 at 6:21 | vote | accept | T. Amdeberhan | ||
| Dec 12, 2018 at 23:15 | comment | added | Zach Teitler | The coefficient of $x^{n-m}$ is the sum of determinants of $m \times m$ principal submatrices of $A_n$. These are skew-symmetric, so have determinant zero when $m$ is odd. When $m=2k$ is even, such a determinant is a square of a Pfaffian, which (roughly speaking) counts weighted perfect matchings of, well, a size $2k$ subgraph of the oriented complete graph with oriented adjacency matrix $A_n$. I wonder if there is any combinatorial way to get $\binom{n}{2k}$ as the total of those squares of counts* of weighted matchings (*but they are not really counts, there are signs...). | |
| Dec 12, 2018 at 10:19 | answer | added | Dima Pasechnik | timeline score: 2 | |
| Dec 12, 2018 at 9:56 | comment | added | Martin Rubey | wouldn't $$\det(M_n)=\sum_{k=0}^m e_{n-2k}(x_1,\dots,x_n)$$ be easier? | |
| Dec 12, 2018 at 8:54 | history | edited | T. Amdeberhan | CC BY-SA 4.0 |
added 1 character in body
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| Dec 12, 2018 at 8:36 | comment | added | Gordon Royle | Do you mean $A_n + x I_n$? | |
| Dec 12, 2018 at 6:33 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |