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Maybe it is a stupid question but I will still ask it here.

How can I prove that the following matrices are not congruent over $\mathbb{R}$?

$A=\begin{pmatrix} 0 & \sqrt10\\ \sqrt10 & -3\\ \end{pmatrix}$ $ B=\begin{pmatrix} 3 & \sqrt6\\ \sqrt6 & -2\\ \end{pmatrix}$

so I found the invertible matrices $P$ and $Q$ such that $P^{t}AP$ and $Q^{t}BQ$ are diagonal matrices. but does it help me ? What am I missing here ? Thanks in advance.

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They are both congruent to the diagonal $(1,-1)$. So they are congruent. Maybe you have a typo? – 1015 Jun 12 '13 at 13:44
@julien Sorry, but I think you are wrong. $A$ is diagonal to (-3,10/3) and $B$ is diagonal to (3,-4) – wantToLearn Jun 12 '13 at 13:46
Yes. Both quadratic forms induced by $A$ and $B$ have the same signature $(1,-1)$. So they are congruent. Maybe what you call congruent is similar via an orthogonal matrix...But congruent is $A=P^TBP$ with $P$ simply invertible. – 1015 Jun 12 '13 at 13:48
See matrix congruence and Sylvester's law of inertia. – 1015 Jun 12 '13 at 13:51

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