Smallest Euclidean norm after matrix multiplication
We can formulate your issue as:
Minimize $\|Ax\|$ under the constraint $\|x\|=1$
than can be re-phrased in the following equivalent way (because it is the Euclidean norm)
Minimize $\|Ax\|^2=(Ax)^T(Ax)=(x^TA^T)(Ax)=x^T(A^TA)x$ under the constraint $\|x\|=1$
As a consequence, it is the eigenvectors of $B:=A^TA$ that must be considered.
(have you heard about SVD and singular vectors and singular values ?)