In mathematics, an invariant subspace of a linear mapping T : V → V i.e. from some vector space V to itself, is a subspace W of V that is preserved by...

In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded...

Lomonosov's invariant subspace theorem is a mathematical theorem from functional analysis concerning the existence of invariant subspaces of a linear...

an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific Journal of Mathematics 16:3 (1966) 421-431 P. Halmos, Invariant subspaces for...

dimensional Euclidean space into invariant subspaces of A. Every Jordan block Ji corresponds to an invariant subspace Xi. Symbolically, we put C n = ⨁...

enough invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace...

invariant subspace of the state space representation of some system is a subspace such that, if the state of the system is initially in the subspace,...

algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under...

years: The basis problem and the approximation problem and later the invariant subspace problem for Banach spaces. In solving these problems, Enflo developed...

linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when...

which proves the invariant subspace claim. In fact, one can conclude something stronger. The range of EB is actually a reducing subspace of A, i.e. its...

Ph.D. thesis advisor was Paul Halmos. His thesis, "On lattices of invariant subspaces" concerns operators on Hilbert space, and most of his subsequent...

Hermiticity,   K n − 1   {\displaystyle \ {\mathcal {K}}^{n-1}\ } is an invariant subspace of A. To see that, consider any   k ∈ K n − 1 {\displaystyle \ k\in...

if every invariant subspace of V has an invariant complement. (That is, if W is an invariant subspace, then there is another invariant subspace P such that...

{\displaystyle G} -invariant subspaces, e.g. the whole vector space V {\displaystyle V} , and {0}). If there is a proper nontrivial invariant subspace, ρ {\displaystyle...

decomposed into the invariant subspaces of A C n = ⨁ i = 1 k Y i . {\displaystyle \mathbf {C} ^{n}=\bigoplus _{i=1}^{k}Y_{i}.} The subspace Yi = Ker(λi − A)m...

Schur decomposition implies that there exists a nested sequence of A-invariant subspaces {0} = V0 ⊂ V1 ⊂ ⋯ ⊂ Vn = Cn, and that there exists an ordered orthonormal...

then the line through 0 and v is an invariant set under T, in which case the eigenvectors span an invariant subspace which is stable under T. When T is...

on a vector space V is semisimple if every T-invariant subspace has a complementary T-invariant subspace. If T is a semisimple linear operator on V, then...

in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace. This is at the level of an observation...

always, invariant manifolds are constructed as a 'perturbation' of an invariant subspace about an equilibrium. In dissipative systems, an invariant manifold...

vector space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace. This is equivalent to the minimal polynomial...

ei(T) T = T ei(T) means the range of each ei(T), denoted by Xi, is an invariant subspace of T. Since ∑ i e i ( T ) = I , {\displaystyle \sum _{i}e_{i}(T)=I...

operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven. General...

W. There is a single word of weight 24, which is a 1-dimensional invariant subspace. M 24 {\displaystyle M_{24}} therefore has an 11-dimensional irreducible...

linear subspace (vector subspace) of a vector space produces an affine subspace of the vector space. One commonly says that this affine subspace has been...

representation is faithful. The subspace C e 2 {\displaystyle \mathbb {C} e_{2}} is a D 6 {\displaystyle D_{6}} –invariant subspace. Thus, there exists a nontrivial...

eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced...

distinct eigenvalues. Any subspace spanned by eigenvectors of T is an invariant subspace of T, and the restriction of T to such a subspace is diagonalizable....

existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the invariant subspace problem. With I. J...