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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! =)

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    $\begingroup$ Why? Because that's the definition of eigenvalue. Can you work out what it means? $\endgroup$ – Najib Idrissi May 27 '13 at 6:01
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Let's view $A$ as a linear operator from vector spaces $V \to W$. An eigenvalue $\lambda$ and eigenvector $x$ satisfy

$$Ax = \lambda x$$

So equivalently:

$$( A - \lambda I ) x = 0$$

But $( A - \lambda I ) 0 = 0$ and $0 \ne x$. This shows that $A - \lambda I$ is not injective.

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\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.

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