TITLE: Injectivity of $A-\lambda I$
QUESTION [9 upvotes]: I'm reading a paper on determinants and on one point the author states that:
A complex number $\lambda$ is called an eigenvalue of matrix $A$ if $A-\lambda I$ is not injective.
Why is this? Could someone clarify :)
Thank you! =)

REPLY [5 votes]: \begin{align*}
A-\lambda I\text{ is not injective} & \iff\exists v_{1},v_{2}\colon v_{1}\neq v_{2}\text{ and }(A-\lambda I)v_{1}=(A-\lambda I)v_{2}\\
 & \iff\exists x\colon x\neq0\text{ and }(A-\lambda I)x=0\\
 & \iff A-\lambda I\text{ is singular}\\
 & \iff\det(A-\lambda I)=0
\end{align*}
The last line regarding the determinant is probably the definition
of eigenvalue you are familiar with.