So far, I have learned matrix norms for constant matrices, e.g.
$$\left\|A\right\|_1 = \max_{1\leq j\leq n}\sum_{i=1}^m\left|a_{ij}\right|$$
and norms of time dependent vectors, e.g.
$$\left\|\mathbf{x}\left(t\right)\right\|_2 = \sqrt{\int_{t_0}^\infty\sum_{i=1}^n\left|x_i\left(t\right)\right|^2}$$
Is there a norm defined for time varying matrices, e.g. $\left\|A\left(t\right)\right\|_1$, $\left\|A\left(t\right)\right\|_2$, and $\left\|A\left(t\right)\right\|_\infty$, whether it be integral or just a "product" e.g. $A^T\left(t\right)A\left(t\right)$?
$A(t)$ is a matrix in terms of $t$. We can define
$$\|A(t)\|_1=\max_{1 \le j \le n}\sum_{i=1}^m|a_{ij}(t)|$$
and it will be a function of $t$. Similarly for the other norms.