# Matrix norm proving problem

I have a homework, one of a few, that I should prove that the norm $||A||_S = \sup\{||Av||:||v||=1\}$ Where ||.|| is some norm on $R^n$ into which vector space belongs also $v$. I should mention that A is a square real or complex matrix with dimension $n$.

Wikipedia says it is a norm, however, first axiom in the definition of norm tells us that $||A||=0$ iff $A=0$. But we can find a matrix, for example: $$\begin{pmatrix}1 & 1\\ 0 & 0\end{pmatrix}$$ and vector $v=(1/\sqrt2, -1/\sqrt2)$ and then according to euclidean norm $||v||=1$ but $||Av||=0$ and clearly A is not zero. What am I missing?

You have to take the sup over all $v$. Take a unit vector orthogonal to yours :)
• You took $v=(1,-1)/\sqrt2$. Take instead $v=(1,1)/\sqrt2$. Indeed, by Cauchy-Schwarz, this $v$ gives you the maximum possible $\|Av\|$. – Ted Shifrin Apr 24 '13 at 23:22
• The $\sup$ means to take the maximum (in this particular case) over all nonzero vectors $v$, not just plug in one vector and call it quits. – hardmath Jan 15 '14 at 13:47
As you have said $||A||$ is defined as the supremum over all possible values of $||Av||$ such that $||v||=1$. For the matrix you have given $||Av||=0$ for the particular vector $(1/\sqrt{2},-1/\sqrt{2})$, however take for instance $v'=(1/\sqrt{2},1/\sqrt{2})$. Then $||Av'||=\sqrt{2}$. Hence the supremum of $||Av||$ aver all vectors such that $||v||=1$ is at least as large as $\sqrt{2}$.