Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 74

We adopt the notation L(V; W) to denote the set of linear transformations from the vector space V to the vector space W.

Definition 4.6 Let and let U be a subspace of V such that where . Then U is called an invariant subspace of V under T, or more briefly, U is T-invariant.

We take some examples. Each subspace is invariant with respect to the following operators:

Let be a basis in and suppose that:

We define the vector :

Then P is a linear operator (called the projection operator) on to the subspace spanned by the vectors .

The projection operator has the following invariant subspaces:

  which remain unchanged and

  that are carried into zero.