# Siegel upper-half space

In dimension 1, the Siegel upper-half space $\mathbb{H}=\{\tau\in\mathbb{C}:\Im\tau>0\}$ has the property that if $\lambda\in\mathbb{C}^\times$, then $a\lambda\in\mathbb{H}$ for some $a\in\mathbb{C}^\times$ (obviously). In higher dimension, if we replace $\mathbb{C}$ by the space of complex symmetric matrices and $\mathbb{H}$ by $\mathbb{H}_g=\{\tau\mbox{ symmetric s.t. }\Im\tau\mbox{ is pos. def.}\}$, this no longer happens: take for example the matrix $$M=\left(\begin{array}{cc}1&\ 0\\0&-1\end{array}\right);$$ there is no complex number $a\in\mathbb{C}^\times$ such that $aM\in\mathbb{H}_g$.

My question is the following: Is there a certain number $r$ such that if $V$ is a complex subspace of symmetric matrices of dimension greater than $r$, then $V$ intersects $\mathbb{H}_g$?

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Hint: The subspace of symmetric matrices with trace $0$ is of codimension $1$ and it does not intersect $\mathbb{H}_g$.