Let $a \in \mathbb R^m$. What is an analytic formula for the maximum value of $\|Xa\|_2$ over all $k \times m$ matrices with $\|X\|_F \le 1$?

Question

Let $a \in \mathbb R^m$.

Question. What is an analytic formula for the maximal value of $\|Xa\|_2$ subject to $X \in \mathbb R^{k \times m}$, $\|X\|_F \le 1$ ?

Trivial ower-bound. Let $r(a)$ be the max value. Observe that $r(a) \ge \|a\|_2$. Indeed, this is the value we get uppon restricting the problem to matrices $k \times m$ matrices $X$ all of whose columns are zero except for the first one.

Answer 1

Using the fact that the operator norm is at most the Frobenius norm, we have

$$\|Xa\|_2 \le \|X\| \cdot \|a\|_2 \le \|X\|_F \cdot \|a\|_2 \le \|a\|_2.$$

You can check that this upper bound is attainable by any matrix of the form $X = \dfrac{1}{\|a\|_2}ua^T$ where $u \in \mathbb{R}^k$ is a unit vector.