Spectral norm of concatenation of two matrices. Let $A \in R^{n*p}$ and $B \in R^{n*d}$ such that both have spectral norm(largest singular value) equal to 1. Now if I concatenate the columns of $A$ and $B$ to construct $M=[A|B]$, then what can we say about the spectral norm of $M$. I am able to bound it by 2 but can we say something stronger(equal to or better upper bound)?

$2$ is the optimum bound, since you can take $n=p=d=1$ and $A=B=1$. Then, in the $\infty$ norm, $\|M\|_\infty = 2$.
