Explicit example of a basis of invertibles for $n\times n$ matrices Using a topological (+linear algebra) argument, one can establish the existence of a basis spanning any square matrix using invertible matrices ( $span(GL_n (\Bbb{R}))=\mathcal{M}_n (\Bbb{R}) $). But I can't seem to find explicit constructions of such bases?
 A: This works for all fields (with characteristic 0 or finite characteristic). Let $E_{ij}$ denotes the matrix with a 1 at the $(i,j)$-th entry and zero elsewhere. Consider the set consisting of


*

*$I$,

*$I+E_{ij}$ for any $i\ne j$,

*$I+E_{ii}+E_{i1}+E_{1i}$ for every $i\ne 1$.


From (1) and (2), one can obtain any $E_{ij}$ with $i\ne j$. Therefore, one can further obtain from (3) any $E_{ii}$ with $i\ne1$. Subtract these $E_{ii}$s from the identity matrix in (1), one obtain also $E_{11}$. Hence every $E_{ij}$ is a linear combination of the $n^2$ matrices in (1)-(3).
A: Such a basis can be constructed by combining three types of matrices:


*

*The $n^2 - n$ matrices that have a $1$ on the diagonal, a $1$ on a single off-diagonal element and are $0$ elsewhere.

*The identity matrix.

*The matrices $\mathrm{diag}(2, 1, \ldots, 1), \ldots, \mathrm{diag}(1, \ldots, 1, 2, 1)$.


More generally, one can construct many bases that only consist of invertible matrices. Simply choose a base $\{I_n, A_2, \ldots, A_{n^2}\}$ and consider the family of matrices $\{I_n, A_2 + \lambda I_n, \ldots, A_{n^2} + \lambda I_n\}$. These vectors are still a base for $\mathbb{R}^{n \times n}$ and are nonsingular if $-\lambda$ is not an eigenvalue of any $A_i$.
