Proving Root Space Invariancy of Linear Transformation

[Sudharaka]

Hi everyone, :)

Here's a question that I don't quite understand.

Given \(f:\, V\rightarrow V\) and a root space \(V(\lambda)\) for \(f\), prove that \(V(\lambda)\) is invariant for \(g:,V\rightarrow V\) such that \(g\) commutes with f.

What I don't understand here is what is meant by root space in the context of a linear transformation. Can somebody please explain this to me or direct me to a link where it's explained?

 

[Opalg]

It looks as though a root space is what I would call an eigenspace, in other words the subspace of all eigenvectors corresponding to a given eigenvalue $\lambda$. (Strictly speaking, an eigenvector has to be nonzero, so the eigenspace is the set of all eigenvectors corresponding to $\lambda$, together with the zero vector.)

 

[Sudharaka]

It looks as though a root space is what I would call an eigenspace, in other words the subspace of all eigenvectors corresponding to a given eigenvalue $\lambda$. (Strictly speaking, an eigenvector has to be nonzero, so the eigenspace is the set of all eigenvectors corresponding to $\lambda$, together with the zero vector.)

Thanks very much for your valuable reply. :) There was some doubt on my mind as to what this root space is all about. It seems to me that there is some ambiguity about this depending on different authors. For example I was reading the following and it has a slightly different definition about the root space.

Matrix And Linear Algebra 2Nd Ed. - Datta - Google Books

However I don't know what my prof. had in his mind when writing down this question. So to make matters simple I shall take the eigenspace as the rootspace. Thanks again, and have a nice day. :)

 

[Opalg]

I was reading the following and it has a slightly different definition about the root space.

Matrix And Linear Algebra 2Nd Ed. - Datta - Google Books

Probably the definition in the book by Datta is the standard one. My previous suggestion was only a guess.

 

[blue_raver22]

Hi there,

Root space in the context of linear transformations refers to the subspace of the vector space V that contains all the eigenvectors corresponding to a particular eigenvalue, denoted by λ. In other words, V(λ) is the set of all vectors v such that f(v) = λv, where f is the linear transformation.

To prove root space invariancy, we need to show that for any vector v in V(λ), g(v) is also in V(λ). In other words, g must map eigenvectors of f with eigenvalue λ to other eigenvectors of f with the same eigenvalue λ.

To do this, we can use the fact that g commutes with f. This means that g(f(v)) = f(g(v)) for any vector v in V. Since v is an eigenvector of f with eigenvalue λ, f(v) = λv. Therefore, g(f(v)) = g(λv) = λg(v). Similarly, f(g(v)) = f(λv) = λf(v) = λ(λv) = λ^2v.

Since g(f(v)) = f(g(v)), we can equate the two expressions and get λg(v) = λ^2v. This shows that g(v) is also an eigenvector of f with eigenvalue λ, and therefore, g(v) is in V(λ). This proves that V(λ) is invariant for g.

I hope this explanation helps. If you need further clarification or have any other questions, please don't hesitate to ask.