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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?

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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.

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