As far as patterns of zeros, I don't know. That sounds like the kind of thing that Dick Stanley must have studied. Victor On Thu, Sep 7, 2017 at 10:53 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Thanks, Victor!
The article looks quite interesting, but also quite tough sledding...
At the topmost level, are there any classes of matrices other than upper triangular, lower triangular, both/diagonal, and/or block versions of same, that preserve the overall pattern of zeros?
From a directed graph adjacency matrix perspective, triangular matrices are partial orders forced into a linear order; block matrices have subsets of vertices with cyclic orders (equivalence classes).
It seems to me that these might exhaust the classes of matrices with patterns of zeros.
At 06:25 AM 9/7/2017, Victor Miller wrote:
Henry, This may be more than you bargained for, but this is a well-studied problem. To make your question more precise, you can ask for algebraic subgroups of matrices -- i.e. subsets of matrices that are the set of solutions to a bunch of polynomials in their coordinates (think determinant = 1, for example). These polynomials are said to "cut out" the group, The following paper surveys the classification of such: https://www.dpmms.cam.ac.uk/~nd332/alg_gps.pdf
Besides the ones that you mentioned, you can also have the set of matrices that preserve a quadratic form: x --> x^T Q x. So you'd want all matrices, A, such that A^T Q A = Q. If A is the identity you get the orthogonal group, If A has dimension 2n and has n +1, and n -1 on the diagonal and 0 everywhere else, you get the Symplectic group.
The nice thing about algebraic groups, is that you can restrict their coordinates to lie in a field which is a subfield of the coefficients used to define the polynomials that cut out the group, you get a subgroup.
Victor
On Wed, Sep 6, 2017 at 9:12 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Is there an encyclopedia/catalog somewhere of classes of square matrices closed under matrix multiply?
I'm interested in generic classes -- e.g., things that work for arbitrary n, e.g., diagonal matrices, orthogonal matrices, upper triangular, lower triangular, etc.
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