Timeline for How could I extend this result to a case where the matrices were not of full rank?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 29, 2020 at 21:27 | comment | added | Kashif | Okay. It looks like they proved the relaxed condition in the paper right before (18). Now the question is how to avoid inverting $B^{-1}$ for the rest of the proof. | |
| Oct 27, 2020 at 10:44 | comment | added | Federico Poloni | @Glassjawed No. Something like the $1\times 1$ case with $A=1$, $B \to 0$, which seems problematic. | |
| Oct 27, 2020 at 10:29 | comment | added | Kashif | Yeah, I think the RHS would have been continuous (continuity of the trace, max of cts function is continuous) if not for the absolute value sign. Is that what you were thinking as well? | |
| Oct 27, 2020 at 10:19 | comment | added | Federico Poloni | @Glassjawed No, sorry, I do not have much time right now and I prefer to interact only through MO (not even in chat). But anyway the argument that A and B are PD on the same subspace looks suspicious to me, thinking about the involved changes of bases. | |
| Oct 27, 2020 at 10:12 | comment | added | Kashif | Hmm...yeah do you perhaps have a few minutes to chat privately? I'm thinking maybe I could use the pseudoinverse $(B'B)^{-1}$ instead of $B^{-1}$ and get a result. But I'm also wondering if the argument that A and B are PD on the same subpsace makes sense to you as well. | |
| Oct 27, 2020 at 10:07 | comment | added | Federico Poloni | @Glassjawed Yes. | |
| Oct 27, 2020 at 10:05 | comment | added | Kashif | LHS and RHS refer to the $\sqrt{tr(A^{1/2}BA^{1/2})}$ and $\max_{X>0} \{ \left|tr X\right|: A\geq XB^{-1}X^{*} \}$? | |
| Oct 27, 2020 at 9:50 | comment | added | Federico Poloni | @Glassjawed Change $A$ and $B$ to $A+\varepsilon I$ and $B+\varepsilon I$, and let $\varepsilon \to 0$. Then clearly the LHS is continuous; the RHS is less clear, at least to me, but I hope something can be said. | |
| Oct 27, 2020 at 8:41 | comment | added | Kashif | Continuity argument in which way? With respect to what? | |
| Oct 27, 2020 at 8:32 | history | answered | Federico Poloni | CC BY-SA 4.0 |