Invariant subspace

In mathematics, an invariant subspace of a linear mapping T : V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W. In mathematics, an invariant subspace of a linear mapping T : V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W. Consider a linear mapping T {displaystyle T} that transforms: An invariant subspace W {displaystyle W} of T {displaystyle T} has the property that all vectors v ∈ W {displaystyle vin W} are transformed by T {displaystyle T} into vectors also contained in W {displaystyle W} . This can be stated as The basis of this uni-dimensional space is simply a vector v {displaystyle v} . Consequently, any vector x ∈ U {displaystyle xin U} can be represented as λ v {displaystyle lambda v} where λ {displaystyle lambda } is a real scalar. If we represent T {displaystyle T} by a matrix A {displaystyle A} then, for U {displaystyle U} to be an invariant subspace it must satisfy: We know that x ∈ U ⇒ x = β v {displaystyle xin URightarrow x=eta v} with β ∈ R {displaystyle eta in mathbb {R} } .