$2$-norm of a non-singular matrix

Question

If I have a non-singular matrix $A$, i.e., $\det(A) \neq 0$, then can I say that surely $\| A \|_2 \neq 0$?

Answer 1

You are right, since $||A||_2=\sqrt{\rho(A^tA)}$, $det(A)\neq 0$ implies $det(A^tA)=det(A)^2\neq0$ and so $A^tA$ does not have $0$ as an eigenvalue. Therefore $||A||_2\neq 0$.

Answer 2

The definition of a norm $|-|$ requires positive-definiteness: $|A|=0$ iff $A=0$. This implies your claim.

To prove positive-definiteness for $||-||_2$ take $A\neq 0$. Now $A$ has a nonzero column $a_i$. It follows $||A||_2\geq|Ae_i|_2=|a_i|_2\neq 0$. It is thus a consequence of positive-definiteness of the euclidean-norm $|-|_2$.