Invariant Subspace Characterization of Triangularability

Theorem

If $T$ is invariant, then:

1. $T$ is upper triangularizable
2. There exists a basis $\{{\alpha }_1,\dots ,{\alpha }_n\}$ for $V$such that if $W_i=span({\alpha }_1,\dots ,{\alpha }_i)$ then:
   • Well Ordered Set, as $W_1\subset W_2\subset \cdots \subset W_n$
   • $T$-Invariant as $T(W_j)\subset W_i,\forall 1\le j\le i\le n$