Question

Why do we impose the requirement that the eigenvector be non-zero when we do not place this requirement on the eigenvalue.

Answer 1

Question 1.

Why do we impose the requirement that the eigenvector be non-zero when we do not place this requirement on the eigenvalue.

Solution. Each eigenvector should have a uniquely defined eigenvalue. Since $A0=0$ for any operator $A$, then we may write that $A0=\lambda 0$ for any scalar $\lambda$. So, any number can be seen as an eigenvalue of the zero vector, if we allow to be an eigenvector. To avoid this ambiguity, an eigenvector is assumed to be nonzero. $\square$