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I am having trouble in understanding eigenvalues and eigenvectors correctly. In particular, why are the eigenvalues of a matrix related to the nullspace of $A-\lambda I$ ? What exactly does it mean?

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Because if $Ax = \lambda x$, then $(A-\lambda I) x = Ax -\lambda x = \lambda x - \lambda x = 0$. This means that $x$ is an eigenvector of $A$ for eigenvalue $\lambda$ if and only if it lies in the nullspace of $A-\lambda I$.

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$\lambda$ is eigenvalue of a matrix $A$ if and only if exists a nonzero vctor $v$ such that $Av=\lambda v=\lambda Iv$ in the other words $(A-\lambda I)v=0 $ thus $A-\lambda I$ has nontrivial null space, Vice versa.

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