Timeline for Spectra of products variously permutated
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 18, 2023 at 20:01 | history | edited | Pietro Majer | CC BY-SA 4.0 |
edited body
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| Dec 18, 2023 at 20:00 | comment | added | Pietro Majer | yes, it is indeed | |
| Dec 18, 2023 at 19:51 | comment | added | Christian Remling | @PietroMajer: I didn't give a $z$ explicitly since it seemed obvious how to finish it (take $z=P_1$ or $z=P_2$). | |
| Dec 18, 2023 at 19:38 | comment | added | Pietro Majer | So this example generalises to any number $n>2$ of factors: if $A_i$ has rank 1 and $A_1e_1=e_ 2, A_2e_2=e_3,\dots, A_ne_n=e_1$, then $A_n A_{n-1}\dots A_2A_1e_1=e_1$, and any product of these factors not in a cyclic permutation is $0$ | |
| Dec 18, 2023 at 19:38 | comment | added | Christophe Leuridan | I corrected the typo. Indeed, three matrices are required to have a counterexample. | |
| Dec 18, 2023 at 19:35 | history | edited | Christophe Leuridan | CC BY-SA 4.0 |
correction of a typo
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| Dec 18, 2023 at 19:27 | comment | added | Christian Remling | There are also counterexamples in $2$ dimensions, for example $x=y^t=\bigl( \begin{smallmatrix} 0 & 1\\ 0 & 0\end{smallmatrix} \bigr)$, so $xy=P_1$, $yx=P_2$. | |
| Dec 18, 2023 at 19:14 | vote | accept | Pietro Majer | ||
| Dec 18, 2023 at 19:09 | comment | added | Pietro Majer | The last line should read $CBAe_j=0$ for $j\in\{2,3\}$, I think (in any case 1 is an eigenvalue by the preceding line) | |
| Dec 18, 2023 at 18:55 | history | answered | Christophe Leuridan | CC BY-SA 4.0 |