Set of Triangular Matrixes

A set of Upper Triangular Matrices, can be represented as:

• $W_1=span(\{{\alpha }_1\})$
• $W_2=span(\{{\alpha }_1,{\alpha }_2\})$
• $W_n=span(\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\})$ They have the following properties:
• Well Ordered Set, as $W_1\subset W_2\subset \cdots \subset W_n$
• $T$-Invariant as $T(W_j)\subset W_i,\forall 1\le j\le i\le n$

Extened in the Invariant Subspace Characterization of Triangularability theorem