On Wikipedia there is a definition of induced norm. In Convex Optimization, Boyd & Vandenberghe extend the induced norm:
As a generalization suppose $\|\cdot\|_a$ and $\|\cdot\|_b$ are norms on $\mathbf R^p$ and $\mathbf R^q$, respectively. The induced norm of a matrix $X \in \mathbf R^{p\times q}$ is defined as $$\|X\|_{a,b}=\sup_{v\ne0}\frac{\|Xv\|_a}{\|v\|_b}.$$ (This reduces to the spectral norm when both norms are Euclidean.) The induced norm can be expressed as \begin{align*} \|X\|_{a,b}&=\sup\{\|Xv\|_a \mid \|v\|_b=1\}\\ &=\sup\{u^TXv \mid \|u\|_{a^*}=1, \|v\|_b=1\} \end{align*} where $\|\cdot\|_{a^*}$ is the dual norm of $\|\cdot\|_a$ and we use the fact that $$\|z\|_a=\sup\{u^Tz \mid \|u\|_{a^*}=1\}.$$
Please explain the meaning of the generalized norm. Why can $a$ and $b$ be different? What is the meaning of each one?
Any help is appreciable. Thanks in advance!
The induced norm tells us the maximum amount that a vector can get stretched out by $X$. But in order for this statement to make sense, we need to have a way of measuring the sizes of vectors in $\mathbb R^p$, and also a way of measuring the sizes of vectors in $\mathbb R^q$. That's what the norms $\| \cdot \|_a$ and $\|\cdot\|_b$ do for us.
There's no reason that we should restrict ourselves to using, say, Euclidean norms.