# Smooth Map between the smooth manifolds $GL_n(\mathbb{R})$ and $S_n(\mathbb{R})$

First of all I am sorry for the bad notation, but I shall denote the smooth manifold of real symmetric $$n \times n$$ matrices by $$S_n(\mathbb{R})$$. I shall denote the transpose of a matrix $$A$$ by $$A^T$$

Define the map $$f:GL_n(\mathbb{R}) \to S_n(\mathbb{R})$$ by $$f(A) \to A^T A$$. I want to show that this map is a smooth map.

I am using the definition from the book An Introduction to Manifolds by Loring W. Tu. For every point $$A \in GL_n(\mathbb{R})$$, I need charts $$(U,U',\phi)$$ and $$(V,V',\psi)$$ such that the map $$\psi \circ f \circ\phi :U' \to V'$$ is smooth. But I am unable to proceed this way. I am unable to get charts and then showing smoothness of the composite function might get complicated.

How should I solve it? Is there some other way for the problem?

• What do charts for these matrices look like? (Hint: Just "flatten" the matrix, this is a chart for it. Restricting the flattening to $GL_n(\mathbb{R})$ and $S_n(\mathbb{R})$ will still provide a chart). In these coordinates, $A^TA$ just has polynomial entries! (Also you need $\psi \circ f \circ \phi^{-1}$ not $\phi$) Commented Apr 10, 2021 at 20:45
• What do you mean by 'flatten the matrix'? Commented Apr 10, 2021 at 21:09
• Use the "obvious" isomorphism between $M_(\mathbb{R})$ and $\mathbb{R}^{n^2}$ Commented Apr 11, 2021 at 1:39

Let $$M_n(\mathbb R) \approx \mathbb R^{n^2}$$ denote the manifold of all $$n\times n$$-matrices and define $$F : M_n(\mathbb R) \to M_n(\mathbb R), F(A) = A^TA .$$ This is a smooth map because each coordinate function $$F_{ij} : M_n(\mathbb R) \to \mathbb R$$ is smooth. Clearly $$F(M_n(\mathbb R)) \subset S_n(\mathbb R)$$. Since $$GL_n(\mathbb R)$$ is open in $$M_n(\mathbb R)$$ and $$S_n(\mathbb R)$$ is a submanifold of $$M_n(\mathbb R)$$, we see that that the restriction $$f : GL_n(\mathbb R) \stackrel{F}{\to} S_n(\mathbb R)$$ is smooth.