Triangulation of Matrices and Linear Maps

Let V be a finite dimensional vector space over the field K, and assume n = dim V ≧ 1. Let A: V → V be a linear map. Let W be a subspace of V. We shall say that W is an invariant subspace of A, or is A-invariant, if A maps W into itself. This means that if w ∈ W, then Aw is also contained in W. We also express this property by writing AW a W. By a fan of A (in V) we shall mean a sequence of subspaces {V₁,..., V _n } such that V _i is contained in V _{i + 1} for each i = 1,... , n - 1, such that dim V _i = i, and finally such that each V _i is A-invariant. We see that the dimensions of the subspaces V₁,..., V _n increases by 1 from one subspace to the next. Furthermore, V = V _n .