What does the space of linear operators on the vector space of $n\times n$ matrices look like?

Question

Is there a complete description of the space of linear operators on $M^{n}(\mathbb{C})$? It must be a space of dimension $n^4$. For example, an operator of the form $A\mapsto MAN$, where $M, N\in M^{n}(\mathbb{C})$ is in fact the Krocker product $M\otimes N$. What about the case of a general linear operator?

Thanks in advance.

Answer 1

Consider two finite collection of matrices $M_1,M_2,\ldots, M_r$, $N_1,N_2,\ldots, N_r$; now take operators defined by you, $A\mapsto M_jAN_j$; their sum is also a linear operator: $A\mapsto \sum_{j=1}^n M_jAN_j$. This process gives you a map $M_n(\mathbf{C})\otimes M_n(\mathbf{C})\to \mathrm{End\,}\big(M_n(\mathbf{C})\big)$. This is an isomorphism. Note that $M_n(\mathbf{C})$ is a simple ring (i.e. no two-sided ideals other than the obvious 2)

Answer 2

The space of linear maps $L: U \to V$ has dimension $(\dim U) (\dim V)$.

In the question $\dim U =\dim V = n^2$, hence the space of such maps has dimension $n^2$.

To see this, pick a basis $u_1,...$ for $U$, and $v_1,...$ for $V$. Then a basis for the linear operators is given by the set of operators $L_{ij}$ defined by $L_{ij} u_k = \delta_{jk} v_i$.