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The set of $n\times n$ real matrices may be notated $\mathbb{R}^{n\times n}$, and the set of square matrices whose entries are binary may be notated $\{0, 1\}^{n\times n}$.

Suppose I have decided that $U^n$ denotes the set of $n\times n$ upper triangular matrices. (Other options appear here.) Is there an elegant way to notate the set of $n\times n$ upper triangular matrices whose entries are binary?

$U^n \cap \{0, 1\}^{n\times n}$ is correct, but inelegant.

$\{0, 1\}^{n(n+1)/2}$ gets the "dimension" right, but not the "dimensions."

I don't think that either $\{0, 1\}^{U^n}$ or $\{0, 1\} \times {U^n}$ are correct. If the former is correct, then by analogy we should have $\{0, 1\}^{n\times n} = \{0, 1\}^{\mathbb{R}^{n\times n}}$. If the latter is correct, then in the $1\times 1$ case we have $U^1 = \mathbb{R}$, and then our set is $\{0, 1\} \times \mathbb{R}$ which is again different.

Max
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1 Answers1

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Note that $U^n$ doesn't capture the fact that the entries are from $\Bbb R$. So, you have to somehow include it in your use of notations. So, the line of notations you were using (i.e., $U^n$) will never give you a way to express the set of all $n\times n$ upper triangular matrices with binary entries.

Now, I don't know of any standard notations as such, but you can take inspiration from the $GL_n(\Bbb R)$ and $SL_n(\Bbb R)$ notations, and use $U_n(\Bbb R)$ for the set of all $n\times n$ upper triangular matrices with entries from $\mathbb R$, and $U_n(\Bbb Z\backslash 2\Bbb Z)$ for the set of all $n\times n$ upper triangular matrices with binary entries.

I hope you find that elegant.

Sayan Dutta
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