Question

Suppose V is finite-dimensional, T belongs to End(V) and lambda in F . Then the

following are equivalent:

(a) lambda is an eigenvalue of T ;

(b) T -lambdaI is not injective;

Answer 1

$\lambda$ is an Eigenvalue of matrrix T iff $(T-\lambda I)X=0$ for some non-zero X.

$\iff X\in ker\{T-\lambda I \} \iff T-\lambda I$ is not injective.

Thus statement a and b are equivalent.