Matrix gain in a direction

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

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

Interpretation as a generalization
An interesting interpretation of matrix gain $$y=Ax$$ comes from approaching it as a generalization of the scalar case $$y=ax$$ : One can interpret the multiplication $$ax$$ as how much $$x$$ increases by multiplication with $$a$$. Similarly, matrix gain aims to find out how much $$x$$ increases by multiplication with $$A$$. However, $$y=Ax$$ is much more interesting because now the gain depends on the direction of $$x$$.

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.