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 $\begin{pmatrix} I_k & &\\ &-I_{r-k} &\\ & & O\end{pmatrix}$.

The trick in situations like this is to first use off-diagonal non-zero entries to produce a non-zero on the diagonal. In your case, adding the third row to the second (and then likewise the third column to the second to preserve symmetry) gives
$$
\begin{pmatrix}
1&0&0&0 \\
0&2&1&0 \\
0&1&0&0 \\
0&0&0&1
\end{pmatrix}
$$
Now you can continue as usual, i.e. by subtracting half of the second row from the third (and likewise for columns):
$$
\begin{pmatrix}
1&0&0&0 \\
0&2&0&0 \\
0&0&-1/2&0 \\
0&0&0&1
\end{pmatrix}
$$
Normalizing (and reordering) the entries gives the signature: $(3,1)$.
