Timeline for Set of Positive Definite matrices with determinant > 1 forms a convex set
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 29, 2013 at 2:29 | vote | accept | Federico Magallanez | ||
| Nov 8, 2013 at 16:45 | history | edited | Federico Magallanez | CC BY-SA 3.0 |
clarify the question
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| Nov 8, 2013 at 16:20 | history | edited | Federico Magallanez | CC BY-SA 3.0 |
fixed typos
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| Nov 8, 2013 at 15:40 | answer | added | Suvrit | timeline score: 11 | |
| Nov 8, 2013 at 15:31 | answer | added | alvarezpaiva | timeline score: 4 | |
| Nov 8, 2013 at 13:51 | comment | added | Federico Magallanez | @FedericoPoloni I don't have much background on this topic, and I don't understand any of closed, smooth, convex, and non-empty interior yet. I'm trying to catch up all the comments / answers so far. | |
| Nov 8, 2013 at 13:48 | comment | added | Joris Bierkens | @Misha: do I understand correctly that you mean simultaneous diagonalization by congruence, as described e.g. in Horn, Johnson - Matrix Analysis, Section 4.5 (in particular Theorem 4.5.15)? | |
| Nov 8, 2013 at 13:41 | comment | added | Federico Poloni | What are your attempts up to now? What is the part where you are stuck? Closed, smooth, convex or non-empty interior? | |
| Nov 8, 2013 at 13:13 | answer | added | Igor Rivin | timeline score: 8 | |
| Nov 8, 2013 at 13:12 | comment | added | Misha | @FedericoMagallanez: This is the unitary diagonalization (not the conjugation); you should think of the quadratic forms defined via the Hermitian matrices and make a simultaneous change of variables to make both forms diagonal. | |
| Nov 8, 2013 at 13:08 | comment | added | ofer zeitouni | write X=aT, Y=(1-a)S where T,S belong to P. Then use the inequality together with a^{1/n}+(1-a)^{1/n}>1 | |
| Nov 8, 2013 at 13:05 | comment | added | Federico Magallanez | @Misha I don't understand your comment. Tow matrices can be simultaneously diagonalizable if and only if they commute. I don't see any reason why two positive definite matrices must commute. Would you please be more specific? | |
| Nov 8, 2013 at 12:49 | comment | added | Misha | Use simultaneous diagonalization to reduce the problem to the case of diagonal matrices. | |
| Nov 8, 2013 at 12:35 | history | asked | Federico Magallanez | CC BY-SA 3.0 |