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Is there a quick way to determine if a $2\times 2$ matrix, $M\in M_2(\mathbb R)$, is congruent to $I_2$ over $\mathbb R, \mathbb C, \mathbb Q$? Without explicitly finding the matrices $P\in M_2$ s.t. $I_2=P^TMP$?

Brainstorm: Perhaps usig ranks and signatures? But if I use that how would I determine over what field is the congruence?

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    Since the matrix $M$ is symmetric or Hermitian its eigenvalues are real and Cholesky is possible at each case. And the law of inertia is valid for Hermitian matrices too.$I$don't know for over the rationals.2011-11-30

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