Topological approach to diagonalization algorithms. arXiv:2204.06111
Preprint, arXiv:2204.06111 [math.AT] (2022).
Summary: In this paper we prove that there exists an asymptotical diagonalization algorithm for a class of sparse Hermitian (or real symmetric) matrices if and only if the matrices become Hessenberg matrices after some permutation of rows and columns. The proof is based on Morse theory, Roberts’ theorem on indifference graphs, toric topology, and computer-based homological calculations.
MSC:
| 57S12 | Toric topology |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 15A20 | Diagonalization, Jordan forms |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37D15 | Morse-Smale systems |
| 55M35 | Finite groups of transformations in algebraic topology (including Smith theory) |
| 05C78 | Graph labelling (graceful graphs, bandwidth, etc.) |
| 57-08 | Computational methods for problems pertaining to manifolds and cell complexes |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
| 55N25 | Homology with local coefficients, equivariant cohomology |
| 55N30 | Sheaf cohomology in algebraic topology |
| 52C45 | Combinatorial complexity of geometric structures |
| 05C62 | Graph representations (geometric and intersection representations, etc.) |
| 05C75 | Structural characterization of families of graphs |
| 57-04 | Software, source code, etc. for problems pertaining to manifolds and cell complexes |
| 55T10 | Serre spectral sequences |
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