Description of characteristic non-hyperinvariant subspaces over the field GF(2)

Abstract

Given a square matrix A, an A-invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T) that commute with A. Shodaʼs Theorem gives a necessary and sufficient condition for the existence of characteristic non-hyperinvariant subspaces for a nilpotent matrix in GF(2). Here we present an explicit construction for all subspaces of this type.Peer ReviewedPostprint (published version

The characteristic subspace lattice of a linear transformation

“…The hyperinvariant subspaces have been characterized in [5] and [2], and the characteristic non-hyperinvariant subspaces in [7]. We recall now both results.…”

“…General conditions for their existence, as well as some examples, can be found in [1,2]. A construction to explicitly obtain all of the characteristic non hyperinvariant subspaces of A is given in [7].…”

“…The paper is organized as follows: In section 2 we introduce the notation and basic results. We present here the structure of the characteristic nonhyperinvariant subspaces of A as obtained in [7]. In section 3 we analyze the properties of the lattice Chinv(A).…”

The lattice of characteristic subspaces of an endomorphism with Jordan–Chevalley decomposition

“…The minimal polynomial of A is p-primary, m A = (x 2 +x+1) 3 , p = x 2 +x+1 separable, and the Jordan-Chevalley decomposition of A is The Segre characteristic of N is (3, 3, 1, 1) with Jordan chains e 1 → e 3 → e 5 → 0 e 2 → e 4 → e 6 → 0 e 7 → 0 e 8 → 0 The characteristic and hyperinvariant subspaces of N are (see [7]):…”

Characteristic subspaces and hyperinvariant frames

“…In [18] a subspace Y is called a minext subspace if it complements a hyperinvariant subspace W such that X = W ⊕ Y is characteristic nonhyperinvariant and X H = W . and D µ = span{u 1 , f u 2 , f 2 u 3 }.…”