= Maximum dimension of a totally isotropic subspace
{title2=$n-(r+|s|)/2$}
For a real <quadratic form> on an $n$-dimensional <vector space>, with rank $r$ and signature $s$ defined as positive minus negative index, the maximum dimension of a <totally isotropic subspace> is $n-(r+|s|)/2$. Quotienting by the <radical of a bilinear form> leaves a nondegenerate form of indices $p,q$. Projection of an isotropic subspace to each sign-coordinate space is injective, bounding its dimension by $\min(p,q)$. The radical and vectors pairing one positive with one negative unit coordinate attain the bound $n-r+\min(p,q)$.
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