# Topology in the set of matrices

Let $M_n(\mathbb{R})$ be the set of real $n\times n$ matrices. I've proved that the map $\left \|\cdot \right \| \mapsto \left \| A \right \| :=\sqrt{\text{tr}(A^tA)}$ is a norm. Then I defined the metric as $d(A,B):=\left \| A-B \right \|$. Now, I want to prove that the orthogonal group $O_n(\mathbb{R})$ is an open set in these metric space:

Let $P\in M_n(\mathbb{R})$. We have to find an $\varepsilon>0$ such that $B(P,\varepsilon)\subset O_n(\mathbb{R})$. Then

\begin{equation*} \begin{split} B(P,\varepsilon)&=\left \{ Q\in O_n({\mathbb{R}})\ |\ \sqrt{\text{tr}(Q^tQ)-2\text{tr}(Q^tP)+\text{tr}(P^tP)}<\varepsilon \right \} \\ &=\left \{ Q\in O_n({\mathbb{R}})\ |\ \sqrt{\text{tr}(I_n)-2\text{tr}(Q^tP)+\text{tr}(I_n)}<\varepsilon \right \}\\ &=\left \{ Q\in O_n({\mathbb{R}})\ |\ \sqrt{n-2\text{tr}(Q^tP)+n}<\varepsilon \right \} \\ &= \left \{ Q\in O_n({\mathbb{R}})\ |\ \sqrt{2}\cdot\sqrt{1-\text{tr}(Q^tP)}<\varepsilon \right \} \end{split} \end{equation*} And now?

• it is quite obvious that in the neighborhood (for your norm) of an orthogonal matrix there is always a non-orthogonal matrix, no ? for example $P+\epsilon I$ is not orthogonal in general. but in general, if $M$ is inversible then $M+\epsilon A$ is inversible when $\epsilon$ is small enough, hence the set of inversible matrix is an open set. – reuns Apr 21 '16 at 19:16

No. You have to look at a set of arbritray matrices in a neighbourhood of a given $P$ and show that any matrix in that set is orthogonal.
(Edit: Actually it is a closed smooth submanifold of lower dimension than $M_n$, so it does not even contain an interior point).
(2nd Edit: maybe you are mixing this up. The General Linear Group $GL_n$ of invertible matrices is a (dense) open subset of $M_n$).
• @Victor This norm is rather common but not very spectacular, it's just the one derived from the standard scalar product in $\mathbb{R}^{n\times n}$, written down using other terms. Any textbook on linear algebra should do. Introductory chapters on manifolds are found in many textbooks on analysis, if you want to have a deep dive you should consult textbooks on differential topology or differential geometry. I'm out of the business since 15 years, so I don't know which textbooks are preferred today. I learned these topics with Hirsch's book on DT and Spivak's comprehensive introduction to DG. – Thomas Apr 21 '16 at 18:02