# Projection onto the kernel of a matrix

by tmatrix
Tags: kernel, matrix, projection
 Emeritus Sci Advisor PF Gold P: 9,224 What do you mean by "projection operator" (or "projector") if not an orthogonal projector? If V is a finite-dimensional vector space and U is a subspace of V, every x in V can be uniquely expressed as x=y+z, with y in U, and z in the orthogonal complement of U. The map $x\mapsto y$ is the projection operator associated with the subspace U. It's linear, self-adjoint and idempotent (P2=P). Let P be any linear, self-adjoint and idempotent operator. Its range W is a subspace. So every x in V can be uniquely expressed as x=y+z, with y in W and z in the orthogonal complement. Since the decomposition is unique, and x=Px+(1-P)x, we have y=Px and z=(1-P)x. So P is the projection operator associated with W. This means that the two standard ways to define a projection operator are equivalent. So if you're using either of these definitions, there's only one projection operator associated with ker M. Are you using some other definition?