Timeline for Singular values of Hadamard Product
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 5, 2018 at 17:24 | answer | added | Igor Rivin | timeline score: 4 | |
| Jun 5, 2018 at 15:56 | comment | added | Jochen Glueck | I see. Apparently, I overlooked your condition on the entries of $A$ and $B$ (still, the scaling behaviour of the inequalities seems strange to me, but this might be due to the fact that I don't have much experience with Hadamard products). Considering your question on whether @fedja's counterexample only occurs on a nullset: By a perturbation argument, the same problem occurs on an open neighbourhood of fedja's matrices (and this neighbourhood intersects the set of matrices which fulfil your condition on the entries in a nonempty open set). | |
| Jun 5, 2018 at 15:47 | comment | added | fedja | I would still be interested in a good upper bound for the first sum. Define "good". In other words, state precisely what quantities you know, in terms of which the estimate should be given. | |
| Jun 5, 2018 at 15:41 | comment | added | Jabor | As i presumed $|A_{i,j}|,|B_{i,j}|<1$ this does not matter since for $t<1$ obviously $t^2<t$ holds. @fedja is right that this does not work for their example, though I wonder if this only happens for some null subset of $\mathbb{R}^{m \times n}$. I would still be interested in a good upper bound for the first sum. | |
| Jun 5, 2018 at 15:13 | comment | added | Jochen Glueck | It seems that the inequalities in your question (both for the singular values and for the Frobenius norm) do not scale correctly: If $A = B = t I$ (where $I$ is the $n\times n$-identity matrix and $t \in [0,\infty)$), then the singular values and the Frobenius norm of $A$ and $B$ grow linearly in $t$, while the singular values and the Frobenius norm of $A \circ B$ grow quadratically in $t$. Maybe you forgot a square root somewhere? | |
| Jun 5, 2018 at 14:10 | review | First posts | |||
| Jun 5, 2018 at 14:17 | |||||
| Jun 5, 2018 at 14:07 | comment | added | fedja | Obviously false even for the norm: take $A=B=\begin{bmatrix}1&1\\-1&1\end{bmatrix}$. Computers spoiled us entirely: we run hundreds of random tests on 10 by 10 instead of thinking for 5 minutes about 2 by 2 :-( | |
| Jun 5, 2018 at 13:59 | history | asked | Jabor | CC BY-SA 4.0 |