# How to represent a Matrix Space?

How can we represent the space of matrices? E.g. A vector $z\in{}R^m$ in the column-space of matrix $A\in{}R^{m\times{}n}$ can be represented as $$z=Ax$$ for some $x$.

Context:

In the following paper, in equation (7), the authors have noted that a particular null-space can be summarized in compact matrix notation as $$Ce_ke^T_k+e_ke^T_kC$$

where $C$ is an arbitrary diagonal matrix (perhaps like $x$ in the example above) and $e_k$ is a vector of length $k$ of all ones. I did not understand how this expression can represent a matrix space.

Blaschko, Matthew B., Wojciech Zaremba, and Arthur Gretton. "Taxonomic Prediction with Tree-Structured Covariances." European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases. 2013.

The expression can describe an space of matrices if you take $C$ to range over all diagonal matrices. As the space of diagonal $n×n$-matrices is naturally isomorphic to $ℝ^n$, you could write the mentioned space of matrices as $$\{Ce_ke_k^T + e_ke_k^TC \mid C \in \mathrm{Diag}(ℝ, k)\} = \{x^T\!e_ke_k^T + e_ke_k^Tx \mid x \in ℝ^n\}$$