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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 the group GL(m, C) x GL(n, C) acting on the space of m x n matrices over C , we introduce a class of subgroups which we call admissible. We suggest an algorithm to reduce an arbitrary matrix to a normal form with respect to an action of any admissible group. This algorithm covers various classification problems, including the "wild problem" of bringing a pair of matrices to normal form by simultaneous similarity. The classical left, right, two-sided and similarity transformations turns out to be admissible. However, the stabilizers of known normal forms (Smith's, Jordan's), generally speaking, are not admissible, and this obstructs inductive steps of our algorithm. This is the reason that we introduce modified normal forms for classical actions.

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