Daniel McLaury

188 reputation

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	188 reputation	bio	website		visits	member for	11 months
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Apr 22	asked	Passing an iterator
Sep 18	awarded	Caucus
Aug 19	accepted	Is there a clean way to extract the subspaces invariant under a list of matrices?
Aug 15	awarded	Nice Question
Jun 8	comment	Is there a clean way to extract the subspaces invariant under a list of matrices? @J.M., thanks for the reference. It actually looks like the paper by Arapura and Peters in its bibliography is much closer to what I want. I'm going to try to implement that now.
Jun 7	comment	Is there a clean way to extract the subspaces invariant under a list of matrices? @J.M., care to elaborate? When I look at the documentation, there seems to be a "with respect to a" option, but no description of what this means or what it does.
Jun 7	comment	Is there a clean way to extract the subspaces invariant under a list of matrices? @DanielLichtblau, C^n is (always) a valid answer. It'd be fine if it was returned as an answer, but it can also be deduced that it's invariant, since it's the direct sum of invariant subspaces already present in the list.
Jun 7	comment	Is there a clean way to extract the subspaces invariant under a list of matrices? @rcollyer, if the matrices are all invertible, then being invariant under the matrices is the same as being invariant under the subgroup of $GL_n$ they generate. Otherwise, they generate a monoid. Alternatively, since being an invariant subspace isn't affected by adding scalar matrices to the $A_i$, you could modify them and reduce the question to the case where they are invertible, though I'm not sure that this accomplishes anything.
Jun 7	comment	Is there a clean way to extract the subspaces invariant under a list of matrices? If they're all scalar multiples of the identity, then we have something of a problem, because then there would be infinitely many distinct invariant subspaces. This could be indicated in a few different ways -- whatever's convenient. (When this happens, it will always be the case that you can just take any subspace of a particular subspace and it will be invariant, so you just return something indicating "This space, together with all its subspaces," either by returning a separate list of these, or tagging them somehow.)
Jun 7	comment	Is there a clean way to extract the subspaces invariant under a list of matrices? It depends on what's on the diagonal. If every diagonal entry is different, and they're 3x3, then the output should be { { { 1 , 0, 0 } }, { { 0, 1, 0 } }, { { 0, 0, 1 } } }. (The list could also contain two- and three- dimensional combinations of these if necessary, although that information would be redundant.)
Jun 7	comment	Is there a clean way to extract the subspaces invariant under a list of matrices? A subspace $W \subseteq V$ is invariant under a matrix $A_i$ if, for each $w \in W$, $A_i w \in W$. I want to find all subspaces invariant under all the matrices simultaneously. In practice, you'd do this by looking at the (generalized) eigenvectors of each matrix (unless there's some nice matrix trick that I'm missing which makes it easier.)
Jun 7	asked	Is there a clean way to extract the subspaces invariant under a list of matrices?
Jun 4	comment	How do I make Mathematica understand roots of unity? Thanks! Upvoted both answers; sad I can only accept one.
Jun 4	awarded	Supporter
Jun 4	awarded	Scholar
Jun 4	awarded	Editor
Jun 4	revised	How do I make Mathematica understand roots of unity? deleted 1 characters in body
Jun 4	accepted	How do I make Mathematica understand roots of unity?
Jun 4	awarded	Student
Jun 4	asked	How do I make Mathematica understand roots of unity?