Which is the canonical base for the Space of Complex matrices $A \in \Bbb{C}^{n\times n}$?
I know in real matrices $\in \Bbb{R}^n$, the canonical base is trivial, just $n$ vectors with a $1$ in the $i$-th component for the $i$-th vector. But for complex matrices, it would be something like a tensor, because we need to generate a complex entry in each component of the vector. Nevertheless, I would like to know which is the formalization of this idea.
I don't know if it is called “canonical”, but a natural basis of $\Bbb C^{n\times n}$ is $\{E_{ij}\mid1\leqslant i,j\leqslant n\}$ wher $E_{ij}$ is the $n\times n$ matrix whise entries are all $0$ except for the entry located at the $i$th line and the $j$th column, which is a $1$.