A method is proposed that allow the reduction of many classification problems of linear algebra to the problem of classifying Hermitian forms. Over the complex, real, and rational numbers classifications are obtained for bilinear forms, pairs of quadratic forms, isometric operators, and selfadjoint operators.Many problems of linear algebra can be formulated as problems of classifying the representations of a quiver. A quiver is, by definition, a directed graph. A representation of the quiver is given (see [6], and also [2,14]) by assigning to each vertex a vector space and to each arrow a linear mapping of the corresponding vector spaces. For example, the quivers 1 e e 1 * * 4 4 2 1 9 9 e e correspond respectively to the problems of classifying:• linear operators (whose solution is the Jordan or Frobenius normal form),• pairs of linear mappings from one space to another (the matrix pencil problem, solved by Kronecker), and

We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskii's algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set C(m,n) of indecomposable canonical m-by-n matrices. Considering C(m,n) as a subset in the affine space of m-by-n matrices, we prove that either C(m,n) consists of a finite number of points and straight lines for every (m,n), or C(m,n) contains a 2-dimensional plane for a certain (m,n).Comment: 59 page

In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying three-valent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated.Comment: 24 page

For a family of linear operators A( λ) : U → U over C that smoothly depend on parameters λ = (λ 1 , . . . , λ k ), V. I. Arnold obtained the simplest normal form of their matrices relative to a smoothly depending on λ change of a basis in U . We solve the same problem for a family of linear operators A( λ) : U → U over R, for a family of pairs of linear mappings A( λ) : U → V, B( λ) : U → V over C and R, and for a family of pairs of counter linear mappings A( λ) : U → V, B( λ) : V → U over C and R.This is the authors' version of a work that was published in Linear Algebra Appl. 302-303 (1999) 45-61.

Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353], based on Sergeichuk's paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous *congruence of pairs of complex Hermitian matrices.