5,100 research outputs found
Characteristic and hyperinvariant subspaces over the field GF(2)
Let be an endomorphism of a vector space over a field . An
-invariant subspace is called hyperinvariant (respectively
characteristic) if is invariant under all endomorphisms (respectively
automorphisms) that commute with . If then all characteristic
subspaces are hyperinvariant. If then there are endomorphisms
with invariant subspaces that are characteristic but not hyperinvariant. In
this paper we give a new proof of a theorem of Shoda, which provides a
necessary and sufficient condition for the existence of characteristic
non-hyperinvariant subspaces.Comment: 18 page
Regular submodules of torsion modules over a discrete valuation domain
A submodule of a p-primary module of bounded order is known to be
regular if and have simultaneous bases. In this paper we derive
necessary and sufficient conditions for regularity of a submodule.Comment: 10 page
Characteristic subspaces and hyperinvariant frames
Let be an endomorphism of a finite dimensional vector space over a
field . An -invariant subspace of is called hyperinvariant
(respectively characteristic) if it is invariant under all endomorphisms
(respectively automorphisms) that commute with . We assume , since
all characteristic subspaces are hyperinvariant if . The
hyperinvariant hull of a subspace of is defined to be the
smallest hyperinvariant subspace of that contains , the hyperinvariant
kernel of is the largest hyperinvariant subspace of that is
contained in , and the pair is the hyperinvariant frame of
. In this paper we study hyperinvariant frames of characteristic
non-hyperinvariant subspaces . We show that all invariant subspaces in the
interval are characteristic. We use this result for the
construction of characteristic non-hyperinvariant subspaces.Comment: 28 page
Linear transformations with characteristic subspaces that are not hyperinvariant
If is an endomorphism of a finite dimensional vector space over a field
then an invariant subspace is called hyperinvariant
(respectively, characteristic) if is invariant under all endomorphisms
(respectively, automorphisms) that commute with . According to Shoda (Math.
Zeit. 31, 611--624, 1930) only if then there exist endomorphisms
with invariant subspaces that are characteristic but not hyperinvariant. In
this paper we obtain a description of the set of all characteristic
non-hyperinvariant subspaces for nilpotent maps with exactly two unrepeated
elementary divisors
Bilinear characterizations of companion matrices
Companion matrices of the second type are characterized by properties that
involve bilinear maps
A class of marked invariant subspaces with an application to algebraic Riccati equations
Invariant subspaces of a matrix are considered which are obtained by
truncation of a Jordan basis of a generalized eigenspace of . We
characterize those subspaces which are independent of the choice of the Jordan
basis. An application to Hamilton matrices and algebraic Riccati equations is
given.Comment: 10 page
Homomorphisms of modules associated with polynomial matrices with infinite elementary divisors
If the inverse of a nonsingular polynomial matrix has a polynomial part
then one can associate with a module over the ring of proper rational
functions, which is related to the structure of at infinity. In this paper
we characterize homomorphisms of such modules.Comment: 10 page
Hyperinvariant subspaces of locally nilpotent linear transformations
A subspace of a vector space over a field is hyperinvariant with
respect to an endomorphism of if it is invariant for all endomorphisms
of that commute with . We assume that is locally nilpotent, that is,
every is annihilated by some power of , and that is an
infinite direct sum of -cyclic subspaces. In this note we describe the
lattice of hyperinvariant subspaces of . We extend results of Fillmore,
Herrero and Longstaff (Linear Algebra Appl. 17 (1977), 125--132) to infinite
dimensional spaces.Comment: 6 page
Hyperinvariant, characteristic and marked subspaces
Let be a finite dimensional vector space over a field and a
-endomorphism of . In this paper we study three types of -invariant
subspaces, namely hyperinvariant subspaces, which are invariant under all
endomorphisms of that commute with , characteristic subspaces, which
remain fixed under all automorphisms of that commute with , and marked
subspaces, which have a Jordan basis (with respect to ) that can be
extended to a Jordan basis of . We show that a subspace is hyperinvariant if
and only if it is characteristic and marked. If has more than two elements
then each characteristic subspace is hyperinvariant.Comment: 13 page
Pairs of Modules over a Principal Ideal Domain
We study pairs of finitely generated modules over a principal ideal domain
and their corresponding matrix representations. We introduce equivalence
relations for such pairs and determine invariants and canonical forms.Comment: 7 page
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