Wild problem

In the mathematical areas of linear algebra and representation theory, a problem is wild if it contains the problem of classifying pairs of square matrices up to simultaneous similarity.^[1][2][3] Examples of wild problems are classifying indecomposable representations of any quiver that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite) Dynkin diagram) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an affine Dynkin diagram).^[4]

Necessary and sufficient conditions have been proposed to check the simultaneously block triangularization and diagonalization of a finite set of matrices under the assumption that each matrix is diagonalizable over the field of the complex numbers.^[5]

See also

• Semi-invariant of a quiver

References

1. Nazarova, L. A. (1974), "Representations of partially ordered sets of infinite type", Funkcional'nyi Analiz i ego Priloženija, 8 (4): 93–94, MR 0354455
2. Gabriel, P.; Nazarova, L. A.; Roĭter, A. V.; Sergeĭchuk, V. V.; Vossieck, D. (1993), "Tame and wild subspace problems", Akademīya Nauk Ukraïni, 45 (3): 313–352, doi:10.1007/BF01061008, MR 1238674, S2CID 122603779
3. Shavarovskiĭ, B. Z. (2004), "On some "tame" and "wild" aspects of the problem of semiscalar equivalence of polynomial matrices", Matematicheskie Zametki, 76 (1): 119–132, doi:10.1023/B:MATN.0000036747.26055.cb, MR 2099848, S2CID 120324840
4. Drozd, Yuriy A.; Golovashchuk, Natalia S.; Zembyk, Vasyl V. (2017), "Representations of nodal algebras of type E", Algebra and Discrete Mathematics, 23 (1): 16–34, hdl:123456789/155928, MR 3634499
5. Mesbahi, Afshin; Haeri, Mohammad (2015), "Conditions on decomposing linear systems with more than one matrix to block triangular or diagonal form", IEEE Transactions on Automatic Control, 60 (1): 233–239, doi:10.1109/TAC.2014.2326292, MR 3299432, S2CID 27053281