Timeline for Why does the variational method simplify in this way when $H$ Hermitian?
Current License: CC BY-SA 4.0
10 events
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Aug 18, 2023 at 2:16 | comment | added | peek-a-boo | Thank you for the nice comment, and yes, that’s exactly it. (a very pedantic correction: “level sets of eigenvectors of $H$” should really be ‘level sets of $\Lambda_1$’:) Also thanks for the edit suggestion. Initially, I intended for the “Extra good to know facts” to be section 1, hence the numbering issue. | |
S Aug 18, 2023 at 2:12 | history | suggested | EE18 | CC BY-SA 4.0 |
I think typo here, as there were only two steps. Changed "have" to "observe" to get across 6 character limit.
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Aug 18, 2023 at 1:09 | comment | added | EE18 | ...a critical point from any pair of eigenvectors with the same eigenvalue. Thus, in studying $\Lambda_1$, and in observing its critical points (the eigenvectors of $H$) while paying attention to the level sets of eigenvectors of $H$, we obtain no more and no less than the critical points of $\Lambda$ if we follow your procedure in the final paragraph of the section. | |
Aug 18, 2023 at 1:08 | comment | added | EE18 | If I may just confirm your conclusion (mostly for myself) in the Putting it all together for Hermitian $H$. step. In the first paragraph of that section, you show that any element of the $\Lambda$ critical points set, call it $C_\Lambda$, then the entries in that pair are in the set of eigenvectors of $H$ and so are critical points of $\Lambda_1$, i.e. in $C_{\Lambda_1}$. Importantly, they also have the same eigenvalues. Conversely, and in your second paragraph of the section, if you have the entire set $C_{\Lambda_1}$, then you can construct... | |
Aug 18, 2023 at 1:04 | comment | added | EE18 | This is exquisite, thank you so much. I completely agree with @TobiasFünke -- I am always assured of a very clear and precise answer when you respond, so thank you for this and for your various other answers to my questions! | |
Aug 18, 2023 at 1:02 | vote | accept | EE18 | ||
Aug 18, 2023 at 0:59 | review | Suggested edits | |||
S Aug 18, 2023 at 2:12 | |||||
Aug 17, 2023 at 22:25 | comment | added | Tobias Fünke | I know comments shouldn't be used for these purposes, but I want to tell you that I really like your posts because of the clear notation! Thank you very much-it is always a pleasure to read and learn from you. | |
Aug 17, 2023 at 22:12 | history | edited | peek-a-boo | CC BY-SA 4.0 |
added 260 characters in body
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Aug 17, 2023 at 22:06 | history | answered | peek-a-boo | CC BY-SA 4.0 |