consider the two quadratic forms Q and P given by
1 Q(x,y,z,w) = $x^2 + y^2 + z^2 + bw^2$
2 P(x,y,z,w) = $x^2 + y^2 + czw$
the source of the question says that P and Q are equivalent over R if b and c are non zero real numbers with b negative ( i would further like to state that quality of source is not reliable ,.,,answers had been wrong in past too . )
i am not able to figure out why do we need that b to be negative ?
consider the matrix from the first quadratic form
\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&b\end{bmatrix}
and the matrix from second quadratic form
\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&\frac{c}{2}\\0&0&\frac{c}{2}&0\end{bmatrix}
i think P and Q must be equivalent over R if b and c are non zero real numbers ?? please correct me if i am wrong
The eigenvalues of the second matrix are $$\{1, \frac{1 \pm \sqrt{c^2 + 1}}{2}\},$$
with $1$ appearing with multiplicity $2$. Therefore, when $c \neq 0$, $P$ has maximal rank and its signature is $(3,1)$. In general $Q$ will be equivalent to $P$ iff both their ranks and signature coincide, so, in this case, iff $b < 0$.