Timeline for On minimal eigenvalue
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Sep 11 at 7:25 | comment | added | Jasmine | Then how would the method work for $M = \mathbf{x}\mathbf{x}^\top + \mathbf{y}\mathbf{y}^\top + \mathbf{z}\mathbf{z}^\top$ for $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^5$ with $\|\mathbf{x}\| = \|\mathbf{y}\| = \|\mathbf{z}\| = 1$? Thanks. | |
| Sep 11 at 6:41 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
A wrong proof og Gen.1 is removed.
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| Sep 11 at 6:38 | comment | added | Ilya Bogdanov | Oh, you are right! So *this * generalization is not proved yet, I will edit the answer. (Anyway, there is no need in commenting at both posts...) | |
| Sep 11 at 2:00 | comment | added | Jasmine | Why did you obtain $$ d_i\geq \lambda_i\cdot \left(\frac{t_i-\lambda_i}{d-2}\right)^{d-2} $$ rather than the opposite direction using the AM-GM inequality? Thanks. | |
| Sep 10 at 20:20 | comment | added | Ilya Bogdanov | That was a typo; now corrected, thanks. | |
| Sep 10 at 20:19 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
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| Sep 10 at 14:35 | comment | added | Jasmine | How did you obtain $\det\Gamma=\det XX^\top$? Thanks. | |
| Sep 9 at 7:57 | comment | added | Ilya Bogdanov | Sorry, I think now you can think a bit by yourself. Here, pne post is usually for one Q+A, not for such a large thread. | |
| Sep 9 at 1:08 | comment | added | Jasmine | What if $ M = \mathbf{x} \mathbf{x}^{\top} + \mathbf{y} \mathbf{y}^{\top} + \mathbf{z} \mathbf{z}^{\top} $ for all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^5$ with $\|\mathbf{x}\| = \|\mathbf{y}\| = \|\mathbf{z}\| = 1$? Thanks. | |
| Sep 8 at 7:02 | comment | added | Jasmine | Thank you for your answer. What if $M = x_1 x_1^\top + x_2 x_2^\top$, for all $x_1, x_2 \in \mathbb{R}^3$ with $\|x_1\| = \|x_2\| = 1$ and $\langle x_1, x_2 \rangle = 0$? Thanks! | |
| Sep 6 at 8:45 | comment | added | Ilya Bogdanov | Upper; sorry, edited. | |
| Sep 6 at 8:44 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
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| Sep 4 at 10:30 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
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| Sep 4 at 7:44 | comment | added | Ilya Bogdanov | I realized that the ikey estimate becomes more transparent by means of the Cauchy--Binet formula, see edits. I have also added an estimate for $\mathbf x_1,\mathbf x_2\in\mathbb R^4$, but I do not know whether it is sharp. | |
| Sep 4 at 7:43 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
The initial estimate is rewritten by means of Cauchy--The secind generalization is added.
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| Sep 3 at 13:16 | comment | added | Ilya Bogdanov | @Jasmine I am unsure of what result would you expect from the method in the general setup… | |
| Aug 31 at 14:26 | comment | added | Ilya Bogdanov | I have added a proof of the $d$-dimensional generalization. | |
| Aug 31 at 14:23 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
A proof of the generalization is added.
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| Aug 31 at 13:07 | history | answered | Ilya Bogdanov | CC BY-SA 4.0 |