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I have to prove that to each of following classes of matrices can be given a manifold structure:

  1. symmetric (denoted with $\mathcal{S}$)
  2. upper triangular
  3. diagonal
  4. null trace.

I am interested in rather simply proofs that do not allow methods from the theory of Lie groups, instead tools like implicit functions theorem, basic results from basic topology, very basic results from manifold theory are allowed.

  • About (1)

I believe that since $\mathcal{S}$ is isomorphic to the upper right triangular matrices, we can consider the bijection $$ M \in \mathcal{S} \mapsto (a_{11}\dots a_{1n},a_{22},\dots,a_{2n},a_{33},\dots a_{nn}) \in \mathbb{R}^{\frac{n(n+1)}{2}}$$ which defines a bijection. So we can trivially induce a topology on $\mathcal{S}$ from the standard topology over $\mathbb{R}^{\frac{n(n+1)}{2}}$, obtaining a single chart $C^\infty$ atlas (please correct if I'm wrong!!!).

  • For (2) and (3)

the reasoning is almost the same as in (1).

  • About (4)

I'd like to try with implicit function theorem, taking the space of $n \times n$ matrices ($n^2$-manifold) as starting point, considering the defining equation $f = \sum_i a_{ii} = 0$ and observing that the gradient of $f$ is not $0$ for each null trace matrix. So we can conclude that the space of null trace matrices is a closed submanifold of $M(n,\mathbb{R})$ of dimension $n^2-1$. Is the preceding reasoning correct? Thanks.

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1), 3), and 4) are vector spaces, and 2) is an affine space; they're all isomorphic as manifolds to $\mathbb{R}^m$ for appropriate $m$. You need very few tools to see this. –  Qiaochu Yuan Nov 15 '11 at 18:58
Indeed 2,3, and 4 are vector subspaces of $\mathfrak{gl}(n,\mathbb{R})$ –  Giuseppe Tortorella Nov 15 '11 at 18:59
For 4), if the trace is 0 then $a_{nn}=-a_{11}-a_{22}-\cdots-a_{n-1,n-1}$. So you only need one chart mapping to $\mathbb{R}^{n^2-1}$ (just "forget" the $a_{nn}$ entry -- it's redundant). –  Bill Cook Nov 15 '11 at 19:08
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1 Answer

All four sets are linear subspaces of $\mathbb R^{n\times n}$, which can be inferred from their description by linear equations in terms of the matrix entries $a_{ij}$:

  1. $a_{ij}-a_{ji}=0$ for $i>j$
  2. $a_{ij}=0$ for $i>j$
  3. $a_{ij}=0$ for $i\ne j$
  4. $a_{11}+\dots +a_{nn}=0$
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