Is there standard notation for the set of diagonal matrices? Specifically if the elements must be nonnegative, i.e. the matrix is positive semi-definite?
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$\begingroup$I'm not sure there is a standard notation per se, but you could construct one using standard notation.
One common way (among others) to specify the set of non-negative reals is $\mathbb{R}_{\ge 0}$. Thus, $\mathbb{R}_{\ge 0}^n$ would be the corresponding Cartesian product (i.e. the set of all nonnegative n-tuples). A standard way to talk about diagonal matrices uses $\text{diag}(\cdot)$ which maps an n-tuple to the corresponding diagonal matrix:
$$\text{diag}:\mathbb{R^n}\rightarrow \mathbb{R^{n\times n}}, \quad \text{diag}(a_1,...,a_n) := \begin{bmatrix}a_1&&\\ &\ddots&\\ &&a_n\end{bmatrix}$$
Thus the set of all positive semi-definite diagonal matrices can be constructed using set comprehension:
$$\{ \text{diag}(v) : v \in\mathbb{R}_{\ge 0}^n \}.$$
If you really want to talk about the elements of this set, it might be more straightforward to define some non-negative tuples $v_a, v_b, v_c,... \in \mathbb{R}_{\ge 0}^n$ first and then talk about $\text{diag}(v_a) , \text{diag}(v_b), \text{diag}(v_c),$ etc. afterwards.
Update:
Using some slight abuse of notation as discussed here, one could simply say that the set of non-negative diagonal matrices is:
$$\text{diag}(\mathbb{R}_{\ge 0}^n).$$
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