Invariant Subspaces and Orthogonality

Question

I'm struggling with a question on invariant subspaces. If someone could possibly help me out here that would be great. The question is as follows:

Let $A ∈ M$ where $M$ is the set of $n \times n$ matrices. Let $U ⊆\Bbb R^n$ be a subspace. Prove that $U$ is $A$-invariant if and only if $U^⟂$ is $A^T$-invariant.

I'm not sure how to approach this question. I know i should start with what we know which is the definition of an invariant subspace which means that if a subspace $U ⊆ V$ is called $T$-invariant if $T(U) ⊆ U$ with $T : V → V$

Thanks a lot in advance

Answer 1

This is an immediate consquence of the fcat that$ \langle A^{T}x, y \rangle =\langle x, Ay \rangle$. For example if $U$ is invariant and $x$ is in $U^{\perp}$ then this equation gives $\langle A^{T}x, y \rangle=0$ for all $y \in U$ because $x$ is orthogonal to $Ay$. Hence $A^{T}x$ belongs to $U^{\perp}$.

Answer 2

Note according to the definition of $A$-invariance that $$U\text{ is }A\text{-invariant}\iff \forall u\in U\quad\exists v\in U\quad v=Au$$i.e. any vector in $U$ is exactly mapped (by $A$) to another vector in $U$ (which intuitively satisfies the definition of invariance). Now we should prove that $$U^{⟂}=\{w\in \Bbb R^n\ | \ w^Tu=0,\forall u\in U\}$$is $A^T$-invariant. Instead, we show that for $w\in U^{⟂}$, $w_1=A^Tw\in U^{⟂}$. Note that $$w^Tu=0\quad , \quad \forall u\in U$$therefore$$w_1^Tu=(A^Tw)^Tu=w^TAu=w^T(Au)=w^Tv=0$$since $u\in U$ implies $v=Au\in U$ and the proof is complete.