norm of vectors u, and v transposed

Question

given vectors $u$ and $v$ in $R^n$ how do i prove that

$||uv^T||_2 = ||u||_2 ||v||_2$

is it sufficient to just calculate?

$||u||_2 = \sqrt{u^2_1+u^2_2} $

$||v||_2 = \sqrt{v^2_1+v^2_2} $

do i just multiply $u$ and $v^T$ and take the vector norm? or do i take the matrix norm here? or am i totally wrong and misunderstood the problem. Thanks in advance

Answer 1

You may begin with the definition of the operator norm: $$ \|uv^T\|_2=\max_{\|w\|_2=1}\|uv^Tw\|_2. $$ Since $v^Tw$ is a scalar, we get $$ \|uv^T\|_2=\max_{\|w\|_2=1}|v^Tw|\|u\|_2 $$ and you may continue from here.

Edit. One may also begin with the equivalent definition that $\|A\|_2=\sqrt{\rho(A^HA)}$. In this case we have $$ \|uv^T\|_2^2=\rho(vu^Tuv^T)=\|u\|_2^2\rho(vv^T) $$ and it remains to prove that $\rho(vv^T)=\|v\|_2^2$. This should be easy if you consider the images of $v$ and $v^\perp$ under $vv^T$.