How do I expand a matrix in terms of a sum of simpler matrices?

Question

Suppose $A$ is a $n\times n$ matrix of numbers, not necessarily of full rank. What methods exist to find a set of simpler matrices in which it's possible to write this matrix as a linear combination? (i) I don't know how to find a suitable set of matrices and (ii) if its possible to find such a set in the case $A$ does not have full rank.

Answer 1

Taking $\mathbb R^{n\times n}$ as a vector space, it is pretty much the same as $\mathbb R^{n^2}$ (they are actually isomorphic). The same methods for finding a basis for $\mathbb R^{n^2}$ work for $\mathbb R^{n\times n}$ and we can "unfurl" basis vectors into basis matrices. As a concrete example, if we think about $2\times 2$ matrices, then we can write any matrix as so: $$ \begin{bmatrix} a & b \\ c & d\end{bmatrix} = a\begin{bmatrix} 1 & 0 \\ 0 & 0\end{bmatrix} + b\begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix} + c \begin{bmatrix} 0 & 0 \\ 1 & 0\end{bmatrix} + d\begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix} .$$ So matrices with only a single nonzero entry with value $1$ form a basis. This does not matter if the matrix has full rank or not.