I am looking for a convenient way to refer to (denote) the space of all real $2 \times 2$ matrices with non-negative determinant.
Is there a standard notation for that?
One approach I played with is $M_2^{\ge 0}$ (assuming $M_2$ is the vector space of all real $2 \times 2$ matrices), but I don't think this is a good choice, as it can be easily confused with the space of positive-semidefinite matrices.
Is the best one can do is to use something like $\operatorname{GL}_2^+ \cup \mathcal{N}$ or $\operatorname{GL}_2^+ \cup \mathcal{Z}$ when $\mathcal{N}$ or $\mathcal{Z}$ denote the space of singular matrices?
(Does that space has a standard notation?)
I find the notation involving the $\cup$ symbol to be a bit cumbersome here. I would prefer to use a "single" symbol. Should I just assign my own symbol for that in a paper?
Any suggestions would be welcomed.
Although not very concise, I think $\{\mathbf{M}\in \mathbb{R}^{n\times n}:\det(\mathbf{M})\geq0 \}$ is the most clear. If you're going to reference this set a lot in your workings, just include something like $$\text{Let } S=\{\mathbf{M}\in \mathbb{R}^{n\times n}:\det(\mathbf{M})\geq0 \}$$ And refer to the set as $S$ (or whatever you choose to use) for the rest of the paper, report, homework, etc.