Timeline for Dimensionality reduction preserving cyclic traces
Current License: CC BY-SA 4.0
11 events
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| Jul 23, 2023 at 19:09 | comment | added | Joseph Van Name | I recently did some computer experiments where I made the following observation. Let $p(x_1,\dots,x_r)$ be a homogeneous non-commutative polynomial of degree $n$ with random coefficients. Let $A_1,\dots,A_r$ be complex polynomials where $\rho(p(A_1,\dots,A_r))^{1/n}/\rho(A_1\otimes\overline{A_1}+\dots+A_r\otimes\overline{A_r})^{1/2}$ is locally maximized. Then $\text{Tr}(q(A_1,\dots,A_r))\approx 0$ whenever $q$ is a homogeneous non-commutative polynomial of degree $m$ and $m\mod n\neq 0$. This means that order $n$ traces do not behave like order $m$ traces for $m\mod n\neq 0$. | |
| Jun 4, 2023 at 17:15 | history | edited | Rodrigo de Azevedo |
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| May 28, 2023 at 23:14 | comment | added | Joseph Van Name | One can probably test to see whether every cyclic function can be written as the trace using general techniques such as using Grobner bases. There are computer algebra systems that can solve this problem when n is small enough and when you give the computer algebra system a particular cyclic function. | |
| May 28, 2023 at 23:00 | history | edited | Joseph Van Name |
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| May 28, 2023 at 22:55 | comment | added | Joseph Van Name | For any particular $n$, the problem is decidable for either the real or the complex numbers. This follows from the reformulation of the question as to whether every cyclic function can be written as $(i,j,k)\mapsto\text{Tr}(A_iA_jA_k)$ which is a first order statement in the theory of real closed fields or respectively algebraically closed fields of characteristic zero. And both the theory of real closed field and algebraically closed fields of characteristic zero are complete and decidable. | |
| May 28, 2023 at 22:33 | comment | added | Joseph Van Name | With the field of complex numbers, we can factorize $B_1,\dots,B_n$ as upper triangular block matrices where the diagonal blocks have no common invariant subspace. We may then apply Burnside's theorem to conclude that each of the diagonal blocks generates the entire matrix algebra. | |
| May 28, 2023 at 22:08 | comment | added | Joseph Van Name | From this question mathoverflow.net/q/391669, we conclude that the functions $(i,j,k)\mapsto\text{Tr}(A_iA_jA_k)$ are precisely the cyclic invariant functions as long as $m$ is large enough. | |
| May 28, 2023 at 16:47 | history | edited | Paul Christiano | CC BY-SA 4.0 |
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| May 28, 2023 at 16:46 | comment | added | Paul Christiano | I'm imagining m >> n, I'll clarify the question. If n = 1, then I have only a single matrix A_1, and I can take B_1 to be the cube root of the trace of the cube (i.e. the l3 norm of the eigenvalues of A). | |
| May 27, 2023 at 14:32 | comment | added | pupshaw | if all the A_i are nilpotent and commute with one another (for example if they are powers of the shift operator in some basis), then all of these products are nilpotent as well and the traces are zero. So we could set all the B_i to zero and your condition would hold. This principle should allow you to build lots of examples of families of matrices where these traces agree. | |
| May 27, 2023 at 1:24 | history | asked | Paul Christiano | CC BY-SA 4.0 |