Find a property of symmetric bilinear forms on finite-dimensional vector spaces above the stated body of the matrices, which keeps the congruent classes of the given symmetric matrices apart from each other.
a)$\begin{bmatrix}1 & 0 & 0\\0 & 1 &0\\0 & 0 & 1\end{bmatrix} and\begin{bmatrix}1 & 0 & 0\\0 & -1 &0\\0 & 0 & -1\end{bmatrix} \epsilon \ Mat(3\times3,{\mathbb{R}})$
b) $\begin{bmatrix}1 & 2 \\2 & 5 \end{bmatrix} and\begin{bmatrix}3 & 5 \\5 & 9\end{bmatrix} \epsilon Mat(2\times2,{\mathbb{Q}})$
c) $\begin{bmatrix}1 & 2 & 2\\2 &4 &4\\2 & 4& 4\end{bmatrix} and\begin{bmatrix}1 & 2 & 2\\2 & 5 &5\\2 & 5 & 5\end{bmatrix} \epsilon \ Mat(3\times3,{\mathbb{R}})$
I know what congruence means and that it is a equivalence relation. However i do not know the properties of the congruence classes, which keep them apart. So far I have noticed, that in
a) the first matrix is positive definite and the second is indefinite
b) The determinant of the first one is 1 and the second one has determinant 2
c) The rank of the 1st is 1 and the second is 2, as the amount of linear dependent vectors is different.
However I do not know if these are criteria s for the different congruent classes. What are those? And where can I find a proof for them?