Describe the orbit structure of the action of $GL_n(\mathbb{R})$ on the space of symmetric matrices

Question

I am reposting this question for better responses after deleting the previous one. I am studying orbit structures in linear algebra and I am having a bit of trouble visualizing the orbit structure of positive definite symmetric matrix $b$ when the group $GL_n(\mathbb{R})$ is acting on the space of symmetric bilinear forms (I know there is a one to one correspondence between symmetric bilinear forms and symmetric matrices so I am going to work with the latter). If the action defined is $S$ -> $A$$S$$A^T$ where $A$∈ $GL_n(\mathbb{R})$ what will be the orbit $GL_n(\mathbb{R})$.$b$ as a set( $b$ is the positive definite symmetric matrix)? Will it turn out to be the space of symmetric matrices ?