Timeline for Jacobi's equality between complementary minors of inverse matrices
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 13, 2019 at 8:24 | comment | added | Alexander Chervov | I feel it is quite similar to what we wrote extending Jacobi identity to matrices with non-commuting entries (Manin matrices) arxiv.org/abs/0901.0235 | |
| Jan 12, 2019 at 21:12 | history | edited | darij grinberg | CC BY-SA 4.0 |
improve latex, update ref, fix typos
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| Dec 19, 2017 at 19:54 | comment | added | darij grinberg | @PietroMajer: This is what I was trying to achieve when I started writing the proof. Unfortunately, it's not directly clear to me how this works. | |
| Dec 19, 2017 at 18:48 | comment | added | Pietro Majer | So the functoriality of $\Lambda^k$ reads as the (generalized) Cauchy-Binet formula: en.wikipedia.org/wiki/Cauchy–Binet_formula#Generalization . In the same spirit, I'd like to see Theorem 1 as a statement on the inverse matrix of the linear map $\Lambda^k(f_A)$, or better, since it also refers to $\tilde I$ and $\tilde J$, of the linear map $\Lambda^k(f_A)^*\sim\Lambda^{n-k}(f_A)$. Can this produce a proof to Theorem 1 consisting in a plain translation of facts about exterior algebra in the language of matrices? | |
| Dec 19, 2017 at 18:48 | comment | added | Pietro Majer | If we agree to see a $\mathbb{K}$-matrix as a function $S\times S\to\mathbb{K}$ , with any finite index set $S$ (not just some $[m]$), then, according to Proposition 3, the matrix of the linear map $\Lambda^k(f_B)$ in the basis $\{e_I\}_{I\in\mathcal{P}_k([n])}$ is $[\det(B^I_J)]_{(I\times J)\in\mathcal{P}_k([n])\times \mathcal{P}_k([n])}$. | |
| Dec 23, 2016 at 22:40 | history | edited | darij grinberg | CC BY-SA 3.0 |
fix typos, update references
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| Aug 4, 2016 at 15:54 | history | edited | darij grinberg | CC BY-SA 3.0 |
corrections
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| Aug 4, 2016 at 15:47 | history | answered | darij grinberg | CC BY-SA 3.0 |