I've been learning about knot theory lately and got stuck on a proof in Cromwells 'Knots and links', where some linear algebra is needed. For anyone interested its on pg. 158.
In the proof he wants to check for invariance for the Alexander-Polynomial under congruence. For congruence he writes:
$det(xP^TMP-x^{-1}(P^TMP)^T)=det(P^T(xM-x^{-1}M^T)P)=det(P^T)det(xM-x^{-1}M^T)det(P)$ $=det(xM-x^{-1}M^T)$
Were $M$ is a Seifert-Matrix and $P$ invertible.
He also writes earlier in the text that any symmetric matrix $A$ with entires in $\mathbb{R}$ is congruent to a diagonal matrix $B$ an that we can find an invertible orthogonal matrix $P$ with $det(P)=\pm 1$ so that $P^TAP=B$.
But in the proof we use any Seifert-Matrix and these don't have to be symmetric, so I don't see why $det(P)$ should be $\pm1$.
What am I missing here?
Thanks for any help.