Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ denotes the transpose of B, and $\sigma(B)$ is the (multi)set of the eigenvalues of $B$.
I'm trying to figure out the tangent space and the normal space to the set
$$S = \{ B \in M_n(\mathbb{R}) \, : \, B^T = B, \sigma(B) = \Lambda\},$$ at the point $A= \text{diag}(\lambda_1,\ldots,\lambda_n)$, the diagonal matrix with diagonal entries $\lambda_i$'s.
Note that $S$ is a manifold in a neighborhood of $A$.
This is my thought process: Define a path in $S$ as follows: $$A(t)= Q(t) \, A \, Q(t)^T,$$ for a family of orthogonal matrices $Q(t)$, such that $Q(0) = I$, the identity matrix. Then $A(0) = A$.
\begin{align*}\dot{A}(t) &= \dot{Q}(t) \, A \, Q(t)\\ &+\, {Q}(t) \, \dot{A} \, Q(t)\\ &+ \, {Q}(t) \, A \, \dot{Q}(t) \end{align*} But $\dot{A} = O$, hence
$$\dot A(0) = \dot Q(0) \, A + A \, \dot Q(0)^T.$$
So, if I know all the tangents/normals to orthogonal matrices at $0$, that is, at $I$, I can describe all the tangents/normals to $S$ at $A$. But it is not too hard to show that the set of all skew-symmetric matrices (i.e. $\{B : B^T = -B\}$) is the tangent space to the orthogonal matrices at $I$. Hence, the tangent space to $S$ at $A$ is $$ T_A(S) = \{ B \, A + A \, B^T : B^T = -B \}.$$
So here are my questions:
Question 0: Is my argument above correct?
Question 1: Why are these all of the tangent vectors? I think(?) $S$ is an $n(n-1)/2$ dimensional manifold, and $T_A(S)$ is $n(n-1)/2$ dimensional, is that really the reason for $T_A(S)$ begin the whole tangent space? Can you elaborate on it, please?
Question 2.1: What is the normal space to the set of orthogonal matrices, at $I$?
Question 2.2: How do you find the normal space to $S$ at $A$? Is it the same as the following? $$\{B \in M_n(\mathbb{R}) : C\, B = O, \forall C \in T_A\}.$$
Edit: Direct calculations yield that $T_A(S)$ is the set of all symmetric matrices with zero diagonals, as long as $\lambda_i$'s are all distinct.
A full explanation from a higher perspective can be found in Li, Rodman and Tsing, Linear operators preserving certain equivalence relations originating in system theory, Linear Algebra and Its Applications, 161:165-225, 1992.