Timeline for Weak majorizations for sum of two hermitian matrices
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2021 at 10:56 | comment | added | Sijie Luo | Ha, I think I have found out a counterexample. Let $A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$, $B=\begin{pmatrix}2&0\\0&0\end{pmatrix}$ and $U=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. It is easy to verify that $\sum_{j=1}^{n}|\lambda_{j}(A+B)|\not=\sum_{j=1}^{n}|\lambda_{j}(U^{*}AU+B)|$. Thank you for your enlightening example! @Chris Ramsey | |
| Dec 1, 2021 at 9:12 | comment | added | Sijie Luo | Many thanks for your wonderful example! And I have a further question: Dose $\sum_{j=1}^{n}|\lambda_{j}(UAU^{*}+B)|=\sum_{j=1}^{n}|\lambda_{j}(A+B)|$ hold true? The example you cnstructed above do satisfy this relation. Here $(\lambda_{j}(C))_{j=1}^{n}$ are eigenvalues of a $n\times n$ hermitian matrix $C$. | |
| Dec 1, 2021 at 9:02 | vote | accept | Sijie Luo | ||
| Nov 30, 2021 at 16:34 | history | answered | Chris Ramsey | CC BY-SA 4.0 |