Timeline for Matrix multiplication and conjugation
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 24, 2016 at 12:55 | vote | accept | Gregory | ||
| Nov 19, 2016 at 3:37 | answer | added | Nathaniel Johnston | timeline score: 1 | |
| Nov 18, 2016 at 16:43 | comment | added | Pushpendre | Also the canonical form of $*-congruence$ may be useful[1], because of the fact that any matrix $V_kBV_k$ that you can create from $B$, you can create from its canonical form, this way you can simplify $B$ and if you study how $A$ interacts with the map that sends $B$ to its canonical map then you may be in business. [1] "Canonical forms for complex matrix congruence and *congruence", by Horn and Sergeichuk. | |
| Nov 18, 2016 at 16:28 | comment | added | Pushpendre | You may already know this, but one way of stating a constrained version of your question is that how well can a map of matrix be approximated by a series of $*-$congruences. because, if you were to constrain yourself, to non singular $V_k$ then $V_kBV_k^*$ is $*-$congruent to $B$ One approach that might be helpful is to consider some property of matrix, that you want to preserve, like inertia. So now you are adding a sequence of matrices that have the same inertia, on the RHS, and then you are comparing the result of the series to $AB$. | |
| Nov 18, 2016 at 13:46 | comment | added | Nathaniel Johnston | I don't know a complete answer, but the summation on the right is an arbitrary completely positive map (if $d \geq n^2$) and thus can send any PSD matrix to any PSD matrix of your choosing. This also means that this infimum is a semidefinite program that can be solved numerically in (for example) MATLAB. Writing down the dual of the semidefinite program would give you some lower bounds on that infimum, if lower bounds are useful to you. | |
| Nov 18, 2016 at 10:24 | comment | added | Gregory | I thought of Hilbert-Schmidt, but any other is fine, if one can say more in that case. | |
| Nov 18, 2016 at 10:12 | comment | added | Ilya Bogdanov | Which matrix norm do you use? | |
| Nov 18, 2016 at 1:27 | history | edited | Gregory | CC BY-SA 3.0 |
added 199 characters in body
|
| Nov 18, 2016 at 1:19 | review | First posts | |||
| Nov 18, 2016 at 1:43 | |||||
| Nov 18, 2016 at 1:18 | history | asked | Gregory | CC BY-SA 3.0 |