Projection (linear Algebra) - Classification

Classification

For simplicity, the underlying vector spaces are assumed to be finite dimensional in this section.

As stated in the introduction, a projection P is a linear transformation that is idempotent, meaning that P2 = P.

Let W be an underlying vector space. Suppose the subspaces U and V are the range and null space of P respectively. Then we have these basic properties:

1. P is the identity operator I on U:
2. We have a direct sum W = U ⊕ V. This means that every vector x may be decomposed uniquely in the manner x = u + v, where u is in U and v is in V. The decomposition is given by

The range and kernel of a projection are complementary, as are P and Q = I − P. The operator Q is also a projection and the range and kernel of P become the kernel and range of Q and vice-versa.

We say P is a projection along V onto U (kernel/range) and Q is a projection along U onto V.

Decomposition of a vector space into direct sums is not unique in general. Therefore, given a subspace V, in general there are many projections whose range (or kernel) is V.

The spectrum of a projection is contained in {0, 1}, as . Only 0 and 1 can be an eigenvalue of a projection. The corresponding eigenspaces are (respectively) the kernel and range of the projection.

If a projection is nontrivial it has minimal polynomial, which factors into distinct roots, and thus P is diagonalizable.