In the mathematical areas of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
and
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, a problem is wild if it contains the problem of classifying pairs of
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
up to simultaneous
similarity. Examples of wild problems are classifying indecomposable representations of any
quiver
A quiver is a container for holding arrows or Crossbow bolt, bolts. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were traditionally made of leath ...
that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite)
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an
affine Dynkin diagram).
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
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not ...
over the field of the complex numbers.
See also
*
Semi-invariant of a quiver
In mathematics, given a quiver (mathematics), quiver ''Q'' with set of vertices ''Q''0 and set of arrows ''Q''1, a representation (mathematics), representation of Q assigns a vector space ''Vi'' to each vertex and a linear map ''V''(''α''): ''V''( ...
References
Linear algebra
Representation theory
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