I want to prove an upper bound for the trace norm of any matrix $W$ which is a linear combination of matrices $M_i \in \mathbb{M}$, $\mathbb{M} = \{uv^T: u \in \mathbb{R}^d, v \in \mathbb{R}^k, \|u \| = 1, \|v\|=1\}$. $\|.\|_*$ is the trace norm.
$$\|W\|_*= \| \sum \theta_iM_i\|_* \le \sum \|\theta_iM_i\|_* \le \sum \theta_i \| M_i\|_* = \sum \theta_i \| uv^T\|_* $$
My question is what is an upper bound for $\| uv^T\|_*$? is it $1$?
Best Answer
For every pair of unit vectors $u,v,$ $\|uv^T\|_* = 1$. In particular, you can say that $$ u \cdot 1 \cdot v^T $$ is a compact SVD.