Projection Operators

Overview

An operator {% P %} over a vector space {% X %} is defined to be a projection operator if

Theorems

• The range and the null space of a projection are disjoint linear subspaces of the vector space {% X %}
• Given two disjoint linear subspaces {% M %} and {% N %} of {% X %} such that {% M \cup N = X %}, there is a projection operator {% P %} such that {% M %} is the range of {% P %} and {% N %} is the null space of {% P %}.
• Given an orthonormal basis {% v_1,...,v_k %} of a subspace of the vector {% X %}
  {% P \vec{x} = \sum_{i=1}^k \langle \vec{x}, \vec{v}_i \rangle \vec{v}_i %}
  is a projection operator into the subspace and {% (\vec{x} - P\vec{x}) \perp \vec{x} %}