Reading a passage from a book I got stuck...
The author says at a certain point:
"Consider a quadratic form in $k$ complex variables $$ Q(z_{1},...,z_{k})=\sum_{i,j=1}^{n}z_{h}z_{j}b_{hj} $$ If we substitute $x^{h}+y^{h}$ for $z^{h}$, and then take the real part of $Q$ we obtain a real quadratic form in $2k$ real variables: $Q'(x_{1},...,x_{k},y_{1},...,y_{k})=$ real part of ($\sum_{i,j=1}^{n}(x_{h}+iy_{h})(x_{j}+iz_{j})b_{hj})$"
Then he wants to prove that
"If $e$ is an eigenvalue of $Q'$ with multiplicity $\mu$, then $-e$ is also an eigenvalue with the same multiplicity $\mu$"
And to prove that he affirms that the identity $$ -Q(z_{1},...,z_{k})=Q(iz_{1},...,iz_{k}) $$ shows that the quadratic form $Q'$ can be transformed into $-Q'$ by an orthogonal change of variables and then concludes what we want to prove follows from that.
I understand why the proof is completed if we show that the quadratic form $Q'$ can be transformed into $-Q'$ by an orthogonal change of variables but it is not clear to me why from the identity above mentioned follows that an orthogonal change of this kind exists.
Thanks for the help