Invariant Subspaces

Defn

We say \(W<V\) is \(T\)-invariant if \(T(W)\subset W\) i.e., \(\forall v\in W, T(v)\in W\)

Let \(W\) be an invariant subspace of \(V\). Characteristic polynomial \(P_{T_W}\) of \(T_W\) divides the characteristic polynomial \(P_T\) of \(T\)

Choose a basis \(\alpha=\{v_1,...,v_k\}\) of \(W\)

Extend this to a basis \(\beta=\{v_1,...,v_k,v_{k+1},...,v_n\}\) of \(V\)

Then \([T]_\beta=\begin{pmatrix}|&|& &|\\ T(v_1)&T(v_1)&...&T(v_1)\\|&|& &|\end{pmatrix}=\begin{pmatrix}A&B\\0&C\end{pmatrix}\)

\(T(v_j)\in W=span(\alpha)\)

\(\forall j\in \{1,...,k\},T(v_j)=\sum ^k_{i=1} a_{i,j}v_i\)

\(A=[T_w]_\alpha\)

\(det([T]_\beta-tI)=det(T-tI)\)

\(P_T(t)=P_{T_W}(t)q(t)\) thus \(P_{T_W}\) divides \(P(T)\)

Let \(W\) be \(T\)-cyclic subspace of \(V\) generated by a non-zero \(v\in V\). Let \(k=dim(V)\), then:

\(\{v,T(v),T^2(v),...,T^{k-1}(v)\}\) is a basis of \(W\)
If \(\sum ^k_{i=0}a_iT^i(v)=0\), then \(P_{T_W}(t)=(-1)^k(\sum ^k_{i=0}a_it^i)\)

Since \(v\neq0\) so \(\{v\}\) is linearly independent

Let \(j\) be the largest integer st \(\beta=\{v,T(v),T^2(v),...T^{j-1}(v)\}\) is linearly independent

Since \(V\) is finite-dimensional; thus, such a \(j\) exists

Suppose \(span(\beta)\) was \(T\)-invariant.

Since \(v\in w\) and \(w\) was T-variant, so \(\beta\subset W\implies span(\beta)\subset W\)

As \(W\) is the smallest \(T\)-variant subspace containing \(v\), so \(W\subset span(\beta)\)

\(\beta\) is a basis of \(W\) so \(j=k\)

To show \(span(\beta)\) is \(T\)-invariant