Given a <matrix> $A$ with <metric signature> containing $m$ positive and $n$ negative entries, the <indefinite orthogonal group> is the set of all matrices that preserve the <matrix representation of a bilinear form>[associated bilinear form], i.e.:
$$
O(m, n) = {O \in M(m + n) | \forall x, y x^T A y = (Ox)^T A (Oy)}
$$
Note that if $A = I$, we just have the standard <dot product>, and that subcase corresponds to the following definition of the <orthogonal group>: <the orthogonal group is the group of all matrices that preserve the dot product>{full}.

As shown at <all indefinite orthogonal groups of matrices of equal metric signature are isomorphic>, due to the <Sylvester's law of inertia>, only the metric signature of $A$ matters. E.g., if we take two different matrices with the same metric signature such as:
$$
\begin{bmatrix}
1 0
0 -1
\end{bmatrix}
$$
and:
$$
\begin{bmatrix}
2 0
0 -3
\end{bmatrix}
$$
both produce <isomorphic> spaces. So it is customary to just always pick the matrix with only +1 and -1 as entries.


Definition of the indefinite orthogonal group