How to convert this symmetric matrix to normal form? I have to find the signature of the symmetric matrix $\begin{pmatrix}1 & 0 & 0 &0\\0 &0 &1 &0\\0&1&0 &0\\0& 0 &0 & 1\end{pmatrix}$. Now to find the signature I want to reduce this matrix to normal form by doing congruence operation.But I am stuck as if we do $R_{23}$ then we have to do $C_{23}$.Then our purpose is not satisfied as we get the same matrix.We should be able to transform this matrix to $\begin{pmatrix} I_k & &\\ &-I_{r-k} &\\ & & O\end{pmatrix}$.

Hint
$$
\left( {\matrix{
   0 & 1  \cr 
   1 & 0  \cr 
 } } \right) = \left( {\matrix{
   { - 1} & 1  \cr 
   1 & 1  \cr 
 } } \right)^{ - 1} \left( {\matrix{
   { - 1} & 0  \cr 
   0 & 1  \cr 
 } } \right)\left( {\matrix{
   { - 1} & 1  \cr 
   1 & 1  \cr 
 } } \right)
$$
