The set of $\lambda$ satisfying $$ (A - \lambda I_n)\vec{x} = 0 $$ is called the eigenspace of a matrix $A$ corresponding to $\lambda$.
Now I want to write eigenspace = the set of eigenvalues $\lambda$ satisfying ... as a set. So I want to use notation like $\lambda_A$ or $E_A$ or even better, $A_{\lambda}$ as candidates for my set but I am not sure. Any suggestions?
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$\begingroup$The set of eigenvalues is not an Eigenspace (set of eigenvectors for a particular eigenvalue, plus $\vec 0$), but rather the spectrum, which you can denote $\sigma_A$.
Your question asks for the set of eigenvalues, but your comment asks for the span of the eigenvectors, which you could call $E_\lambda$ or $E_\lambda(A)$ as in the Cliff's Notes page Amzoti linked.
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