Projection onto the kernel of a matrix

In summary, the projection operator P is unique if the vector space is finite dimensional, and there is a standard way to refer to it among mathematically knowledgeable people.
  • #1
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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?
 
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  • #2
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 [itex]x\mapsto y[/itex] 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?
 
  • #3
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!
 

What is the definition of "Projection onto the kernel of a matrix"?

The projection onto the kernel of a matrix is a mathematical operation that projects a vector onto the subspace spanned by the nullspace of the matrix. In other words, it is the process of finding the closest vector in the nullspace of a matrix to a given vector.

Why is "Projection onto the kernel of a matrix" important in linear algebra?

The projection onto the kernel of a matrix is important in linear algebra because it helps us understand the structure and properties of a matrix. It also has many practical applications, such as in data analysis, image processing, and machine learning.

How is "Projection onto the kernel of a matrix" calculated?

The projection onto the kernel of a matrix is calculated using the projection matrix, which is the matrix that performs the projection operation. The projection matrix is obtained by first finding the basis for the nullspace of the matrix and then using it to construct the projection matrix.

What is the difference between "Projection onto the kernel of a matrix" and "Projection onto the column space of a matrix"?

The projection onto the kernel of a matrix and the projection onto the column space of a matrix are two different operations. While the projection onto the kernel finds the closest vector in the nullspace of a matrix to a given vector, the projection onto the column space finds the closest vector in the column space of a matrix to a given vector.

What are some real-world examples of "Projection onto the kernel of a matrix"?

One real-world example of projection onto the kernel of a matrix is in image processing, where it is used to remove noise from images. Another example is in linear regression, where it is used to find the best-fit line for a set of data points. It is also commonly used in signal processing and data compression.

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