Timeline for Eigenvalues in unit disk for a 2×2 block matrix
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jan 18, 2023 at 15:16 | comment | added | MathMax | @FedericoPoloni Exactly, I suspected that the lack of symmetry could trigger counterexamples. In fact, non-symmetric matrices can get arbitrarily far from being positive definite even if all their eigenvalues are positive, thereby jeopardising my line of proof. Your nice counterexample is exactly what I was looking for. | |
| Jan 18, 2023 at 15:12 | history | edited | LSpice | CC BY-SA 4.0 |
Tidying, especially `\label`+`\eqref`, while this is on the front page
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| Jan 18, 2023 at 15:09 | comment | added | Federico Poloni | Ah OK I have seen the edit, you are assuming that $A$ is symmetric, too. That should settle the contradiction. | |
| Jan 18, 2023 at 15:07 | comment | added | Federico Poloni | I am still trying to figure out why exactly, but one of us must be wrong because my answer contains a counterexample with $C$ symmetric positive definite and $B$ symmetric. If you can solve the dilemma let me know. :) | |
| Jan 18, 2023 at 15:05 | comment | added | MathMax | @FedericoPoloni The purpose of my post was to give a partial answer, by addressing the special case when B is symmetric (and also A as per my latest edit). Hope this is sufficient to revert the downvote. | |
| Jan 18, 2023 at 14:58 | history | edited | MathMax | CC BY-SA 4.0 |
Clarified that part 3 of the proof holds true only when A and B are both symmetric
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| Jan 18, 2023 at 14:47 | comment | added | MathMax | @FedericoPoloni Because $(\lambda I - B)^{-1}$ and $C$ are (i) both symmetric (since $B$ and $C$ are symmetric by assumptions) and (ii) they commute. | |
| Jan 18, 2023 at 10:19 | comment | added | Federico Poloni | Why is $(\lambda I - B)^{-1}C$ symmetric? | |
| Jan 17, 2023 at 21:33 | history | edited | MathMax | CC BY-SA 4.0 |
Splitted equation (A2) for better readability
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| Jan 17, 2023 at 21:26 | history | edited | MathMax | CC BY-SA 4.0 |
Fixed typos
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| May 26, 2021 at 12:24 | history | edited | MathMax | CC BY-SA 4.0 |
Simplified proof of part 1
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| S May 26, 2021 at 12:14 | history | suggested | Ivano | CC BY-SA 4.0 |
Correction of a typo
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| May 26, 2021 at 12:13 | review | Suggested edits | |||
| S May 26, 2021 at 12:14 | |||||
| May 23, 2021 at 22:55 | history | edited | MathMax | CC BY-SA 4.0 |
Fixed typo
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| May 23, 2021 at 15:41 | history | edited | MathMax | CC BY-SA 4.0 |
improved notation
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| May 23, 2021 at 15:29 | history | edited | MathMax | CC BY-SA 4.0 |
Added proof of last statement.
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| May 22, 2021 at 7:26 | history | edited | MathMax | CC BY-SA 4.0 |
Simplified proof of part 2
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| May 21, 2021 at 14:17 | history | edited | MathMax | CC BY-SA 4.0 |
Added new to step towards complete solution
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| May 21, 2021 at 13:09 | history | answered | MathMax | CC BY-SA 4.0 |