# normal form of antisymmetric matrices under pseudo-orthogonal transformations

It is well-known that any real anti-symmetric $n \times n$ matrix $A$ can be transformed via
$A \to O A O^T$ into block-diagonal form consisting of $2 \times 2$ antisymmetric matrices, where $O \in SO(n)$ is orthogonal.

It seems that the analogous statement should also hold for $O \in SO(p,n-p)$, or at least for $SO(1,n-1)$. What is the precise statement, and where can one find it?

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What do you mean by "analogous statement"? I mean, do you take $O\in SO(p,n-p)$ and $A$ in the Lie algebra $\mathfrak so(p,n-p)$? – Claudio Gorodski Apr 2 '12 at 0:58
I mean $O \in SO(p,n-p)$ and $A$ a real anti-symmetric matrix. Then $O A O^T$ is also anti-symmetric, and there should be a normal form, presumably in terms of block-diagonal matrix consisting of $2 \times 2$ antisymmetric matrices. – Harold Steinacker Apr 3 '12 at 15:00