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I would like to know if there exists a notation for the set of symmetric matrices and symmetric positive definite matrices. For instance, the set of $N \times N$ matrices with real entries is denoted as $\mathbb{R}^{N \times N}$.

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    $\begingroup$ There are lots of them. It depends on who you read. Some people write $Sym_n(\mathbb{R})$ for the symmetric case and $P^+_n (\mathbb{R})$ for the SPD case. $\endgroup$
    – Wintermute
    Oct 30, 2013 at 21:49
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    $\begingroup$ There are lots of them [as there are lots of people called John Smith:-)], there's no "standardised" notation. $\endgroup$ Oct 30, 2013 at 22:06

2 Answers 2

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There is no single "canonical" notation for these. Books tend to not introduce notation for matrix sets that are not closed under matrix multiplication. In articles, people come up with various notations, such as:

  • $\operatorname{Sym}_n$, $\operatorname{Sym}_n(\mathbb R)$, $S^n(\mathbb R)$, $S\mathbb R^{n\times n}$ for symmetric matrices
  • $PSD_n$, $P_n^+$, $S^n_+ $ for positive semidefinite matrices
  • $S^n_{++}$ for positive definite matrices.

The list is certainly not exhaustive.

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From Stephen Boyd & Lieven Vandenberghe's Convex Optimization:


Convex Optimization, Stephen Boyd and Lieven Vandenberghe

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