Matrix gain in a direction

The gain of a matrix $$A \in \mathbb{R}^{m\times n}$$ in a direction $$x$$ is defined as the ratio:

$$\frac{||Ax||}{||x||}$$

What's the point of Matrix gain?
One of the most important applications of matrix gain is to tell us what's the direction of highest gain. In fact, this is defined as the matrix norm. In words, the matrix norm tells us how large we can make $$||Au||$$ for a unit vector $$||u||$$. This, for example, would be the the principal component in the context of PCA.