Timeline for Is trace of a slice of an elementary function of a matrix also elementary?
Current License: CC BY-SA 4.0
15 events
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Mar 29, 2021 at 17:39 | comment | added | M.G. | @Anixx: yup. You should probably ask it in a different question since it's a completely different beast. | |
Mar 29, 2021 at 17:38 | comment | added | Anixx | Okay, so the infinite-dimensional case remains | |
Mar 29, 2021 at 17:38 | vote | accept | Anixx | ||
Mar 29, 2021 at 17:36 | vote | accept | Anixx | ||
Mar 29, 2021 at 17:37 | |||||
Mar 29, 2021 at 17:34 | comment | added | M.G. | A small roller-coaster for you :-) I missed originally that $W$ is actually a fixed constant matrix, in which case the link I posted previously does complete the finite-dimensional case via a small modification, since $x$ is just a scalar (see the 2nd edit). | |
Mar 29, 2021 at 17:32 | history | edited | M.G. | CC BY-SA 4.0 |
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Mar 29, 2021 at 17:18 | comment | added | M.G. | @Anixx: the first edit was incorrect, so I removed it. The link computes the matrix value at a point only. It is not a priori clear to me how the Jordan Normal Form depends on the matrix coefficients, e.g. continuously, smoothly, holomorphically, via elementary functions etc. in the general finite-dimensional case. But this has probably been already investigated somewhere. Nevertheless the link en.wikipedia.org/wiki/Analytic_function_of_a_matrix contains an explicit formula for the 2x2 case. Apologies about the mishap. | |
Mar 29, 2021 at 17:16 | history | edited | M.G. | CC BY-SA 4.0 |
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Mar 29, 2021 at 17:12 | comment | added | M.G. | @Anixx: no, it only completes my comment about the Jordan Normal Form in the finite-dimensional setting. | |
Mar 29, 2021 at 17:10 | comment | added | Anixx | Does your EDIT1 include the infinite-dimensional case? | |
Mar 29, 2021 at 17:04 | history | edited | M.G. | CC BY-SA 4.0 |
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Mar 29, 2021 at 16:52 | comment | added | Anixx | Just what I observe: there is such element $W$ that $\operatorname{reg}\log (W+x)=\psi(x)$, where $\operatorname{reg}$ is extracting the scalar part. | |
Mar 29, 2021 at 16:50 | comment | added | M.G. | The infinite-dimensional setting is certainly a very interesting case, but there are obviously many issues to be considered. For starters, since you care about the trace, you have to restrict to trace-class operators / nuclear operators. I cannot offer you any insight there at the moment. | |
Mar 29, 2021 at 16:44 | comment | added | Anixx | What about an infinite-dimensional case? I am bearing in mind a commutative algebra where it is not the case (although it is not strictly matrix algebra). | |
Mar 29, 2021 at 16:40 | history | answered | M.G. | CC BY-SA 4.0 |