# Projection onto the kernel of a matrix

tmatrix
If we have a matrix M with a kernel, in many cases there exists a projection operator P onto the kernel of M satisfying [P,M]=0. It seems to me that this projector does not in general need to be an orthogonal projector, but it is probably unique if it exists. My question: is there a standard name for such a projector among math people?

Staff Emeritus
Gold Member
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?

tmatrix
Dear Fredrik,
Thank you for the reply. I think I was not sufficiently clear about the concept I am considering.

I want to consider a linear operator M on a vector space V whose image is linearly independent from its Kernel.

On a finite dimensional vector space, this implies that V = ker M + Img M.
In this case, there is a unique projector P such that ker P = Img M and Img P = ker M. It can be considered the natural projector onto the kernel of M. It is not necessarily an orthogonal projector---note that I have not specified any notion of inner product on V.

If the vector space is infinite dimensional, in general we do not have
V = ker M +Img M. But suppose that ker M is a closed subspace of V, so there is a projection P onto the kernel of M: ker M = Img P. If we in addition require P M =0, this is a natural infinite dimensional analogue of the projection operator defined in the last paragraph.

My question is twofold:
1) Is the projection operator defined above unique in the infinite dimensional case?
2) Is there a standard way to refer to this projection operator among mathematically knowledgeable?

Thank you!