Timeline for Eigenvalues of the matrix obtained by letting some of the rows vanish, hoping for some inequality
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 1, 2020 at 4:27 | comment | added | Dima Pasechnik | as you drop rows, you might consider left eigenvectors rather than right ones. eigenvalues are the same for the left and for the right ones. | |
| Dec 31, 2019 at 17:23 | comment | added | Federico Poloni | There is a reference in my link, which should explain the main result. "Interlacing inequalities" in my comment referred to the theorem in the answer there, which applies to submatrices, not to Weyl's theorem on $A+B$. The fact that those two matrices have the same eigenvalues is easy to prove; just use math.stackexchange.com/questions/522385/… on $A-xI$. | |
| Dec 31, 2019 at 16:43 | comment | added | Learning math | @FedericoPoloni Thanks for your comment. Perhaps I'm not aware of the relevant results: could you please provide some relevant references? P.S. I'm aware of Wey's interlacing inequality - en.wikipedia.org/wiki/Weyl%27s_inequality, which bounds the eigenvalues of the sum of two matrices, but not sure how that applies to my case? Also, I'm not taking principal minor when I'm building $A_k$ out of $A$, I'm dropping a few rows (not the corresponding columns as well). How to prove that the principal minor and $A_k$ have same egenvalues? I'd appreciate a more detailed explanation. | |
| Dec 31, 2019 at 16:36 | comment | added | Federico Poloni | If you zero out some rows of $A$, you get (up to permutation) $\begin{bmatrix}A_{11} & A_{12} \\ 0 & 0\end{bmatrix}$, which has the same nonzero eigenvalues as the principal submatrix $A_{11}$. This is a classical setup; there are interlacing inequalities between a matrix and its principal submatrix; see for instance math.stackexchange.com/questions/1670000/… . | |
| Dec 31, 2019 at 16:23 | history | edited | Learning math | CC BY-SA 4.0 |
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| Dec 31, 2019 at 15:51 | history | asked | Learning math | CC BY-SA 4.0 |