Quadratic form.

Question

Consider the quadratic form $Q(v)=v^{t}Av,v=(x,y,z,w)$ where matrix $A$ is given by \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\\ \end{bmatrix}

Then which of the following is true?

1. $Q$ has rank 3.

2. $xy+z^{2}=Q(Pv)$ for some invertible real matrix $P.$

3. $xy+y^{2}+z^{2}=Q(Pv)$ for some real invertible matrix $P.$

4. $x^{2}+y^{2}-zw=Q(Pv)$ for some some real invertible matrix $P.$

It is clear that $Q$ has rank $4$ so $1$ is not true. How about other options. Please help me . Thanks in advance.

Answer 1

You have to find the matrices that generate the forms on the left hand sides. For example, $$ xy + z^2 = \left\langle\left(\begin{matrix}0 & \frac 1 2 & 0 & 0\\\frac 1 2 & 0 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 0\end{matrix}\right)\left(\begin{matrix}x \\ y \\ z \\ w\end{matrix}\right),\left(\begin{matrix}x \\ y \\ z \\ w\end{matrix}\right)\right\rangle. $$ But this matrix has rank 3. Thus, no way.