# Injectivity of $A-\lambda I$

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! =)

• Why? Because that's the definition of eigenvalue. Can you work out what it means? – Najib Idrissi May 27 '13 at 6:01

Let's view $A$ as a linear operator from vector spaces $V \to W$. An eigenvalue $\lambda$ and eigenvector $x$ satisfy
$$Ax = \lambda x$$
$$( A - \lambda I ) x = 0$$
But $( A - \lambda I ) 0 = 0$ and $0 \ne x$. This shows that $A - \lambda I$ is not injective.