What is the topology on the set of symmetric matrices?

Asked Sep 13 '22 at 08:40

Active Sep 14 '22 at 10:07

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What is the topology on the set of $n\times n$ real symmetric matrices?

So, it is claimed that the set of nonnegative definite matrices are closed, but what counts as an open set (topology) in this case?

(Kind of bonkers I can't find an answer to this elementary question by googling).

edited Sep 14 '22 at 10:07

asked Sep 13 '22 at 08:40

det

5

It's the subspace topology with respect to the usual Euclidean topology on $\mathbb{R}^{n \times n}$. So a limit is just an entrywise limit. It's very common to implicitly take the subspace topology like this. – Qiaochu Yuan Sep 13 '22 at 08:42
Yeah, thanks. I see. – det Sep 13 '22 at 08:45
It's a metric space where the distance between $A$ and $B$ is defined to by $\operatorname{Tr}((A-B)^T(A-B))$ – Akiva Weinberger Sep 13 '22 at 12:23
Take a look at this – Rodrigo de Azevedo Sep 14 '22 at 10:09

0 Answers0