flag

A flag is a nested tower of subspaces, each strictly inside the next: {0} = V0 inside V1 inside V2 inside ... inside Vn = V, where each step adds exactly one dimension when the flag is complete. Picture a sequence of ever-larger nested rooms, the smallest a point, the largest the whole space.

The flag is T-invariant if every floor of the tower is an invariant subspace: T(Vi) is contained in Vi for all i. This single nested chain is the geometric content of triangular form. Choose a basis so that the first j vectors span Vj; then because T pushes Vj into Vj, the matrix of T has no entries below the diagonal — it is upper-triangular.

So triangularizing an operator and finding a complete invariant flag are the same act, just phrased in matrix versus subspace language. This viewpoint generalizes beautifully: in Lie theory the set of all complete flags is the flag variety, and Borel subgroups are exactly the stabilizers of flags. The diagonal entries of the triangular matrix are the eigenvalues of T acting on the successive one-dimensional quotients Vi / V(i-1).