Abstract
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 {V 1,..., 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 V 1,..., V n increases by 1 from one subspace to the next. Furthermore, V = V n .
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© 1987 Springer Science+Business Media New York
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Lang, S. (1987). Triangulation of Matrices and Linear Maps. In: Linear Algebra. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1949-9_10
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DOI: https://doi.org/10.1007/978-1-4757-1949-9_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3081-1
Online ISBN: 978-1-4757-1949-9
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