Timeline for Waring's problem for matrices
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Apr 10, 2010 at 9:47 | comment | added | Gerhard Paseman | On a related topic, someone posted a problem in sci.math.research about 10 years ago: What matrices are expressible as the sum of invertible matrices? That problem solves easily, but there is an interesting twist: what matrices are expressible as the sum of exactly two invertible matrices? David Radcliffe and I found only one exception among all nxn matrices over many rings R where R belongs to a large class of rings that includes all fields. This result might interest the original poster. Gerhard "Ask Me About System Design" Paseman, 2010.04.10 | |
| Apr 10, 2010 at 2:29 | comment | added | JSE | One remark: Waring's problem usually takes place in the context of integers, not real numbers. So one might ask: what can one say about sums of squares of nxn matrices? Here there is a nice story,having to do with Siegel modular forms and powers of the Siegel theta function; see e.g. A.N.Andrianov's book "Quadratic forms and Hecke operators." | |
| Apr 10, 2010 at 1:25 | comment | added | Yemon Choi | @Steve: two still sounds suspicious. Every matrix (in fact, every element of a C*-algebra) is a linear combination of four unitaries, by a standard trick. Perhaps the wikipedia entry is assuming that the matrix is self-adjoint? (the proof of the 4-unitary result goes via this special case) | |
| Apr 10, 2010 at 1:22 | comment | added | Steve Huntsman | It's fixed now. | |
| Apr 10, 2010 at 0:46 | comment | added | François G. Dorais | Wikipedia missed the requirement that the norm of the matrix should be ≤ 1. | |
| Apr 10, 2010 at 0:28 | comment | added | Bjorn Poonen | That sounds wrong to me. The set of unitary n by n matrices is compact, so the set of averages of two unitary matrices is compact too. So the latter set cannot equal the space of all n by n matrices. | |
| Apr 10, 2010 at 0:28 | vote | accept | Portland | ||
| Apr 10, 2010 at 0:27 | answer | added | Bjorn Poonen | timeline score: 14 | |
| Apr 9, 2010 at 23:56 | comment | added | Steve Huntsman | According to Wikipedia, every matrix is the average of two unitaries (en.wikipedia.org/wiki/Unitary_matrix#Properties). | |
| Apr 9, 2010 at 23:45 | history | asked | Portland | CC BY-SA 2.5 |