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On normal forms for the similarity classes of matrices and pairs of matrices

Klaus Bongartz

Abstract

We answer two questions posed 1998 in the book 'Arnolds problems'. First, over any field k there is a representative system for the similarity classes of nxn-matrices which is a finite disjoint union of affine subspaces. And second, for n>1 an analogous statement fails for pairs of nxn-matrices over any algebraically closed field of characteristic 0.

On normal forms for the similarity classes of matrices and pairs of matrices

Abstract

We answer two questions posed 1998 in the book 'Arnolds problems'. First, over any field k there is a representative system for the similarity classes of nxn-matrices which is a finite disjoint union of affine subspaces. And second, for n>1 an analogous statement fails for pairs of nxn-matrices over any algebraically closed field of characteristic 0.

Paper Structure

This paper contains 9 theorems, 16 equations.

Key Result

Theorem 1

Each matrix $A \in k^{n\times n}$ is similar over $k$ to exactly one rational normal form $R(A)=R(P_{1},P_{2},\ldots ,P_{r})$. Here $R(A)$ as well as an invertible matrix $T$ with $T^{-1}AT=R$ are obtained from the coefficients of $A$ by finitely many rational operations the number of which grows po

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Theorem 2
  • Definition 4
  • Lemma 2
  • Theorem 3
  • Proposition 1
  • ...and 3 more