Proposition 30.1.2.label Let $n \in \natp$ and $x \in M_{n}(\complex)$, then $\sigma(x)$ is the set of eigenvalues of $x$.
Proof. By the rank-nullity theorem, $\lambda - x$ is invertible if and only if $\lambda$ is not an eigenvalue of $x$.$\square$
Proposition 30.1.2.label Let $n \in \natp$ and $x \in M_{n}(\complex)$, then $\sigma(x)$ is the set of eigenvalues of $x$.
Proof. By the rank-nullity theorem, $\lambda - x$ is invertible if and only if $\lambda$ is not an eigenvalue of $x$.$\square$
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