On Seymour’s Decomposition Theorem

On Seymour’s Decomposition Theorem

Let M be a class of matroids closed under minors and isomorphism. Let M be a 3-connected matroid in M with an N-minor and let N have an exact k-separation (A, B). If there exists a k-separation (X ,Y ) of M such that A ⊆ X and B ⊆ Y , we say the k-separation (A, B) of N is induced in M. In this paper we give new sufficient conditions to determine if an exact k-separation of N is induced inM.

On the Krohn-Rhodes Cascaded Decomposition Theorem

The Krohn-Rhodes theorem states that any deterministic automaton is a homomorphic image of a cascade of very simple automata which realize either resets or permutations. Moreover, if the automaton is counter-free, only reset automata are needed. In this paper we give a very constructive proof of a variant of this theorem due to Eilenberg.

The Cyclic Decomposition Theorem

Let V be a finite-dimensional F -vector space, and let T : V → V be a linear transformation. In this note we prove that V is a direct sum of cyclic T -invariant subspaces. More specifically, we prove that V is a direct sum of cyclic T -invariant subspaces whose annihilators are generated by powers of irreducible polynomials, and that the collection of these polynomials is uniquely determined. R...

The cyclic decomposition theorem

8 Matrices 16 8.1 Representation of vector spaces and linear transformations . . . . . . . . . . . . . . . 16 8.2 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Direct sums of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.4 The companion matrix of a polynomial . . . . . . . . . . . . . . . . . . . . . . . ....

A Martingale Decomposition Theorem