For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset

$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid rk{A}=2r \}$$

is a manifold of dimension $2n(2r)-(2r)^2$ (see this question).

If I now take a look on Hermitian $n\times n$ matrices of rank $2r$ with exact $r$ positive and $r$ negative eigenvalues, how can I determine the dimension of this manifold?

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset

$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$

is a manifold of dimension $2n(2r)-(2r)^2$ (see this question).

If I now take a look at Hermitian $n \times n$ matrices of rank $2r$ with exactly $r$ positive and $r$ negative eigenvalues, how can I determine the dimension of this manifold?

For the vectorspace $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset

$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid rk{A}=2r \}$$

is a manifold of dimension $2n(2r)-(2r)^2$ (see this question: https://math.stackexchange.com/questions/518202/what-is-the-codimension-of-matrices-of-rank-r-as-a-manifold).

If I now take a look on hermitian $n\times n$ matrices of rank $2r$ with exact $r$ positive and $r$ negative eigenvalues how can I determine the dimension of this manifold?

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset

$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid rk{A}=2r \}$$

is a manifold of dimension $2n(2r)-(2r)^2$ (see this question).

If I now take a look on Hermitian $n\times n$ matrices of rank $2r$ with exact $r$ positive and $r$ negative eigenvalues, how can I determine the dimension of this manifold?