The special orthogonal group $SO(n)$ is the subgroup of the special linear group $SL(n)$ of $n\times n$ matrices with determinant one that preserve a non-degenerate symmetric bilinear form. If such a bilinear form is taken to be the one associated to the identity matrix, then $$SO(n):=\{M\in SL(n)\: | \: MM^{t} = I\}$$ Is there a nice description of the group preserving a singular symmetric bilinear form? For instance, if we take $I_{k}=(a_{i,j})$ to be the matrix with $a_{i,i} = 1$ for $1\leq i\leq k$, $a_{i,i} = 0$ for $k+1\leq i\leq n$, and $a_{i,j} = 0$ for $i\neq j$, then how can we describe the group $$SO_{I_k}(n):=\{M\in SL(n)\: | \: MI_kM^{t} = I_k\}$$ of determinant one matrices preserving $I_k$?
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