Timeline for $A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 19, 2020 at 23:38 | comment | added | Robert Israel | Slightly more generally, suppose $A$ is a $2 \times 2$ positive definite symmetric matrix. The partial derivative of $(A^{-1})_{1,1}$ with respect to the entry $a_{1,2}$ is $2 a_{12} a_{22}/(\det A)^2$. Thus if $a_{12} > 0$, increasing $a_{12}$ (and keeping $a_{21}=a_{12}$) increases $(A^{-1})_{11}$ rather than decreasing it. | |
| Oct 19, 2020 at 18:32 | vote | accept | user812951 | ||
| Oct 19, 2020 at 16:24 | comment | added | Carlo Beenakker | sorry, no idea why this inequality should hold in some general sense. | |
| Oct 19, 2020 at 16:15 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 192 characters in body
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| Oct 19, 2020 at 15:54 | comment | added | user812951 | Can I take the liberty to ask you here itself, will the inequality hold true if we put any more conditions on A? Even a hint would be enough. | |
| Oct 19, 2020 at 14:30 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |