# Matrix times eigenspace associated to its eigenvalue

Let $$A$$ be a square matrix of dimension $$n$$ and let $$λ ∈ C$$ be an element of the spectrum of $$A$$. $$X_λ$$ denotes the eigenspace of $$A$$ associated to $$λ$$;In which case do we have $$AX_λ = \{0\}$$? The solution is $$\lambda=0$$. However it is not clear to me what the product between the matrix A and the eigenspace of $$\lambda$$ actually means, given that $$X_λ$$ is actually a set of vectors. What is the product between a matrix and a set of vectors?

In this case you should think of $$AX_\lambda$$ as $$\{A \mathbf x : \mathbf x \in X_\lambda\}$$, ie the set of vectors obtained by multiplying every vector in $$X_\lambda$$ with $$A$$.
You see this kind of shorthand used quite often in maths - where $$f$$ is some function defined on a set $$S$$, $$f(S)$$ can denote the image of $$f$$. For example, $$2\mathbb Z$$ is sometimes used to mean the even integers.