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?