Let $v=(3 ,1 ,3 ,-4)$ and $A=v^t\cdot v$. Find the eigenvalues and eigenvectors of $A$ without calculating the whole matrix $A$
$Rank(AB) \leq min((Rank(A),Rank(B))$ so $Rank(A)\leq 1$ but because $A$ isn't the zero matrix $Rank(A)\neq 0$ and therefore $Rank(A)=1$.
$A$ is $4 \times 4$ size matrix so by rank-nullity theorem we get $4 = 1 + Null(A)$ $\rightarrow$ $Null(A)=3$ so $A$ is not invertible and $0$ is one of its eigenvalues.
How can I find the other eigenvalues and eigenvectors?