Suppose the following non-square matrices, $W \in \mathbb{R}^{1 \times 6}$ and $B \in \mathbb{R}^{6 \times 2}$ and a vector $u \in \mathbb{R}^{2 \times 1}$.
What kind of upper bound can I put on the following multiplication;
$$
WBu,
$$
where $u$ has the following bounds: $u_{\text{min}} \leq u \leq u_{\text{max}}$. For a quadratic form ($u^TWBu$), I know this can be upper bounded by $\lambda_{\text{max}} (WB) \|u\|^2$, where $\lambda_{\text{max}}$ is the largest eigenvalue of $WB$ if it is a semi positive definite matrix. For this non quadratic form I'm looking for a similar upper bound
Matrix $B$ is known and has no special properties. Matrix $W$ is not known and is free to be chosen.
(Personally, I do not come from a strong mathematical background, my apologies for any trivial writing mistakes.)
Best Answer
One such upper bound is $$ WBu \leq \|WB\| \cdot \|u\|, $$ where $\|WB\|$ denotes the "induced 2-norm" of $WB$, which in this case is simply equal to the norm of the row-vector $WB$. More general, $\|M\|$ is the maximal singular value of $M$.