What is the $1$-norm of a matrix?

Question

Let $A \in \mathbb C^{n \times n}$ be some matrix. I'm trying to understand what $\lvert\!\lvert A \rvert\!\rvert_1$ means. I found two definitions:

1. $\lvert\!\lvert A \rvert\!\rvert_1$ is the Schatten norm for $p=1$, so $\lvert\!\lvert A \rvert\!\rvert_1 := \mathrm{Tr}\left(\sqrt{A^\dagger A}\right)$.

2. $\lvert\!\lvert A \rvert\!\rvert_1$ is the maximum of the absolute column sums.

Are these two definitions equivalent? If so, how does one see that?

Answer 1

No, these are two different norms as you can easily see with a randomly generated example.

If you're working with Schatten $p$-norms, please take the time to explain your notation in your writing.

If the only Schatten norm that you're using is $p=1$, then a commonly used notation is $\| A \|_{*}$.