Timeline for Relate existence of common invariant subspace of $m$ matrices to reducibility of certain polynomial
Current License: CC BY-SA 4.0
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| Dec 16, 2020 at 7:22 | comment | added | Xueqing Wen | @GeoffRobinson After I add the comment above, I realize that I am not so sure about the "If the matrix (aij) above is similar to a blocked upper trianglar matrix, then we would have a subalgebra" part | |
| Dec 16, 2020 at 7:19 | comment | added | Xueqing Wen | @GeoffRobinson Consider the algebra $\mathscr{A}$ generated by $\{A_i\}$, then the vector space $K^n$ is an $\mathscr{A}$ module. So if we have a nontrivila subalgebra of $\mathscr{A}$, then we would have a submodule of $K^n$, which is a common invariant subspace of $\{A_i\}$. If the matrix $(a_{ij})$ above is similar to a blocked upper trianglar matrix, then we would have a subalgebra. So the question reduces to(which is somehow my original question): if I have a $n\times n$ matrix over a UFD, and its determinant is reducible, is it similar to a blocked upper trianglar matrix? | |
| Dec 15, 2020 at 10:00 | comment | added | Geoff Robinson | I suppose it carries some information, but I am not sure how much. | |
| Dec 15, 2020 at 4:57 | comment | added | Xueqing Wen | @Geoff Robinson Thanks for comment! I read about the group determinant and I realize that the group determinant can be "defined" for any finite dimensional algebra(in my case I would consider the algebra generated by $A_i$) in the following way: find a linear basis $e_1, \cdots ,e_n$, form a matrix with entry $a_{ij}=e_ie_j$(and then express in terms of linear combination of $\{e_k\}$). If I regard $e_i$ as variables, then can the determinant of $(a_{ij})$ tell something? | |
| Dec 14, 2020 at 16:37 | comment | added | Geoff Robinson | As a particular case of your question, you might like to read about the group determinant. When the $A_{i}$ form a group, the group determinant goes a long way to answering your question. | |
| S Dec 14, 2020 at 15:27 | history | suggested | RobPratt | CC BY-SA 4.0 |
Corrected spelling in title
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| Dec 14, 2020 at 14:27 | review | Suggested edits | |||
| S Dec 14, 2020 at 15:27 | |||||
| Dec 14, 2020 at 9:10 | review | First posts | |||
| Dec 14, 2020 at 10:13 | |||||
| Dec 14, 2020 at 9:06 | history | asked | Xueqing Wen | CC BY-SA 4.0 |