Timeline for Sufficient conditions to order the solutions to a system of linear equations
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 30, 2023 at 13:48 | comment | added | menritgs | That's true, maybe what I meant was a property that would be easier to check than checking each component of $A^{-1}d$. | |
| Mar 30, 2023 at 8:23 | comment | added | Dima Pasechnik | But I have given you a property - a formula (a system of polynomial inequalities in entries of $A$ and $d$) which has to hold true for $x_1>x_2...$. Just as the usual answer to "what is a property for quadratic polynomial $f(x)=ax^2+2bx+c$ to have in order to have no real roots" is given by $b^2<ac$. | |
| Mar 30, 2023 at 8:02 | comment | added | menritgs | I guess I was hoping for results like 'if matrix $A$ and vector $d$ satisfy property $X$ or is of type $T$, then $x_1>x_2>\dots$', but perhaps no such result exists at this level of genearlity. In any case, thanks again for your suggestions. | |
| Mar 30, 2023 at 8:01 | comment | added | menritgs | Yes, that's right, edited, thanks. | |
| Mar 30, 2023 at 0:00 | comment | added | Dima Pasechnik | it's certainly a property of A and d, not A or d. | |
| Mar 29, 2023 at 9:51 | comment | added | Dima Pasechnik | you can use en.wikipedia.org/wiki/Cramer%27s_rule - which would amount to basically the same thing. | |
| Mar 29, 2023 at 9:31 | comment | added | Dima Pasechnik | There is nothing left to "recover" left here, the inequalities can be read off $A^{-1}d$. One further simplification is that instead of $A^{-1}$ you can use the classical adjoint (en.wikipedia.org/wiki/Adjugate_matrix) - this would allow cubic inequalities (quadratic in $a_{ij}$'s, linear in $d_k$'s - but then you'd need to consider two cases, depending on the sign of $\det A$. | |
| Mar 29, 2023 at 8:49 | comment | added | menritgs | Thanks, I agree, though would you know of any results that give conditions on $A^{-1}d$ such that we may recover these type of inequalities? Or maybe some related results? | |
| Mar 28, 2023 at 22:58 | comment | added | Dima Pasechnik | it will suffice for the matrix $A=(a_{ij})$ to be invertible and the coordinates of the vector $A^{-1}d$ to satisfy your inequalities | |
| Mar 28, 2023 at 20:36 | history | edited | menritgs | CC BY-SA 4.0 |
added 5 characters in body
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| S Mar 28, 2023 at 20:07 | review | First questions | |||
| Mar 28, 2023 at 22:58 | |||||
| S Mar 28, 2023 at 20:07 | history | asked | menritgs | CC BY-SA 4.0 |