Timeline for On closest unitary matrix
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| May 9, 2015 at 11:26 | history | edited | Suvrit | CC BY-SA 3.0 |
added disclaimer
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| May 9, 2015 at 11:20 | comment | added | Suvrit | Ok, I think this needs some more thought then! | |
| May 9, 2015 at 7:39 | comment | added | Omid Hatami | @Suvrit: In the case of my example above we have $\geq$. | |
| May 8, 2015 at 22:20 | comment | added | Omid Hatami | I have a counterexample. Let A=1 and $B=\cos \theta$. Then \begin{equation*} V = \begin{pmatrix} \cos \theta & \sin \theta\\ \sin\theta & -\cos \theta \end{pmatrix} \end{equation*} Then \begin{equation*} \left\|\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \begin{pmatrix} \cos \theta & \sin \theta\\ \sin\theta & -\cos \theta \end{pmatrix} - I\right\| = \left\| \begin{pmatrix} \cos \theta - 1 & \sin \theta\\ -\sin\theta & \cos \theta - 1 \end{pmatrix}\right\| \neq \left\| \begin{pmatrix} \cos \theta - 1 & 0 \\ 0 & \cos \theta - 1 \end{pmatrix}\right\| \end{equation*} | |
| May 8, 2015 at 21:40 | comment | added | Omid Hatami | Can you tell me why this identity is true? \begin{equation*} \left\|\begin{pmatrix} A & 0\\ 0 & -A^* \end{pmatrix}V - I\right\| = \|(AB-I)\oplus (A^*B^*-I)\|. \end{equation*} | |
| May 8, 2015 at 21:28 | vote | accept | Omid Hatami | ||
| May 8, 2015 at 22:23 | |||||
| May 8, 2015 at 15:54 | comment | added | Hans | Thank you, Suvrit. Now I realized that you were referring the application of Lidski's theorem to the Lemma, rather than the main body of the proof. I have found that reference. | |
| May 7, 2015 at 1:09 | comment | added | Suvrit | @Hans: the link that I put in the Lemma contains the entire proof; it is also based on block matrices, and ends up reducing to the case where we can use Lidskii's majorization: $$\lambda^{\downarrow}(X)-\lambda^{\downarrow}(Y) \prec \lambda(X-Y) \prec \lambda^{\downarrow}(X)-\lambda^{\uparrow}(Y)$$ for Hermitian matrices. If you are unable to locate the reference, I can try to supply details of that proof also as a part of my answer. | |
| May 7, 2015 at 0:50 | comment | added | Hans | Would you be so kind as to expand on your previous comment (now deleted) on the proof essentially follows from the Lidski Inequality? I do not see a sum of Hermitian matrices here, whereas Lidski Inequality seems to be a property of the eigenvalues of the sum of two Hermitian matrices. | |
| May 7, 2015 at 0:42 | comment | added | Hans | I got the construction for my second question by seeking a symmetric solution for the *'s in your first version of $V$ which turns out to be exactly the answer you provided. I really appreciate you adding the details, and the references. | |
| May 6, 2015 at 20:59 | comment | added | Suvrit | @Hans: I added the requested details. | |
| May 6, 2015 at 20:55 | history | edited | Suvrit | CC BY-SA 3.0 |
added the details requested by Hans
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| May 6, 2015 at 7:08 | comment | added | Hans | Could you please explicate in your answer above how the first inequality is obtain, especially why it is true for every unitary matrix $V$, and why the contraction B could be dilated to a unitary matrix? Thank you. | |
| May 5, 2015 at 17:46 | vote | accept | Omid Hatami | ||
| May 8, 2015 at 21:01 | |||||
| May 5, 2015 at 2:06 | history | edited | Suvrit | CC BY-SA 3.0 |
added some detail; fixed some typos.
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| May 4, 2015 at 23:45 | history | answered | Suvrit | CC BY-SA 3.0 |