Invariant subspace of a representation
From Encyclopedia of Mathematics
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of a group (algebra, ring, semi-group)
in a vector space (or topological vector space)
A vector (respectively, a closed vector) subspace such that for any
and any
one has
. If
is a projection operator from
onto
, then
is an invariant subspace of
if and only if
for all
. The subspace
in
is invariant for any representation in
; it is called the trivial invariant subspace; the remaining invariant subspaces (if there are any) are called non-trivial. See also Contraction of a representation; Completely-reducible set; Irreducible representation.
How to Cite This Entry:
Invariant subspace of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_subspace_of_a_representation&oldid=12674
Invariant subspace of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_subspace_of_a_representation&oldid=12674
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article