Test whether the foloowing quadratic forms are Equivalent

Question

Consider two quadratic forms:

$Q(x,y,z,w)=x^{2}+y^{2}+z^{2}+bw^{2}$ and

$P(x,y,z,w)=x^{2}+y^{2}+czw$.

For what type of values of $b$ & $c$ (real or complex or negative or positive or zero) $P$ & $Q$ are equivalent?

For 2 or 3 variables we can do it easily using some theorems (If $Q$ is non-degenerate quadratic form in $n-variables$over $\mathbb{C}$ then $Q$ is equivalent to $x_{1}^{2}+...+x_{n}^{2}$.) or finding $rank$ , $discrimanant$, $etc$.

But for 4 variables how we can proceed? Are there any general rule to show two quadratic forms equivalent for arbitrary $n-variables$?

Answer 1

The question amounts to finding when the matrices $$ A = \pmatrix{1\\&1\\&&1\\&&&b}, \quad B = \pmatrix{1\\&1\\&&0&c/2\\&&c/2&0} $$ are congruent.

The answer in this case will be that the forms are equivalent iff $b < 0$ and $c \neq 0$.

Answer 2

This may be useful http://en.wikipedia.org/wiki/Witt%27s_theorem#Witt.27s_cancellation_theorem

Reduce it to $z^2 + b\, w^2$ and $c \,z w$ are equivalent.