Suppose $M_r$ is the set of all real symmetric matrices of order $n$ with rank $r$.
(a) Show that $M_r$ is a submanifold of the space $\mathbb R^{n^2}$.
(b) Find the tangent space of $M_r$ at some $A\in M_r$.Suppose $N_G$ is the set of all real symmetric matrices of order $n$ whose zero-nonzero patterns give a fixed graph $G$.
(a) Show that $N_G$ is a submanifold of the space $\mathbb R^{n^2}$.
(b) Find the tangent space of $N_G$ at some $A\in N_G$.
I know the answers. But I don't know how to get there using traditional regular value as given in the following theorem:
Theorem.(Guillemin-Pollack, p 21 & 24) Let $y\in Y$ be a regular value of a smooth map $f:X\to Y$ and $Z=f^{-1}(y)$. Let $x\in Z$. Then
(a) $Z$ is a submanifold of $X$ with $\dim Z=\dim X-\dim Y$, and
(b) the tangent space $T_x(Z)$ of $Z$ at $x$ is the kernel of $df_x:T_x(X)\to T_y(Z)$.