Where $\|u\|_2=\|v\|_2=1$. I think we cannot use $\|uv^*\|_2\leq \|u\|_2\|v^*\|_2=1.$ So are there any other methods?
2026-04-23 15:09:11.1776956951
Can we prove that $\|uv^*\|_2 \leq 1$?
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For any vector $w$ (of the same dimension as $u$ and $v$), we have $$ ||uv^*w||_2=||(v^*w)u||_2=|v^*w|\leq ||v||_2||w||_2=||w||_2 $$ using the homogeneity of the norm, and then Cauchy-Schwarz.
Therefore $||uv^*||_2\leq 1$, and taking $w=v$ shows that in fact $||uv^*||_2=1$.